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^ntkft-36nnfe 

OP 

MECHANICS 


AND 


Engineering. 

CONTAINING A MEMORANDUM pF FACTS 
AND CONNECTION 

OF 

Practice and Theory. 

BY 


JOHN W. NYSTROM, C. E. 

»♦ 


Third Edition Revised, with additional matter* 

* 

















Entered according to Act of Congress in the year 1S56, by 
JOHN W. NYSTROM, 

in the clerk’s Office of the District Court of the United States in and for the 
Eastern District of Pennsylvania. 


.(to^fll bflB 




















PREFACE. 


Let every Engineer make his own Pocket-Book as he 
proceeds in study and practice, it will then suit his 
particular business ; the present work was compiled in 
this way during a professional career of ten } T ears. It 
was not originally intended for publication, but had 
grown so large in manuscript as to be inconvenient; 
this circumstance, combined with repeated requests to 
publish it, has lately placed it before the public 

The Author hopes the introduction of Algebraical 
formulas ins'tead of written rules will be favourably 
received ; Written rules adopted by English Authors 
and Engineers are indeed excellent;—but formulas are 
better,—because, they not only tell what is to be done, 
but at a glance, impress the mind with the complete 
operation. If all the formulas in this beck were 
explained in words, it would he far too large for the 
pocket. 

It is not necessary to understand Algebra for the 
use of the formulas,—011I3' practise (he insertion of 
numerical values, and perform the arithmetical ‘ opera¬ 
tions indicated by the particular formula used. The 
Author lias furnished the fornuffas ready to receive what 
is given, and'refund what is required. 






Navigation. 


r 


ADVERTISEMENT. 


The undersigned is prepared to furnish Drawings and estimates for Pro¬ 
peller Steamers. He designs Engines and Propellers suitable for any 
desired description of Vessels. Furnishes Drawings of Vessels, with their 
whole internal arrangements distinctly shown in sections and details of any 
desired Scale. 

Several years of experience in this profession enable him to furnish com¬ 
plete and correct Drawings on very short notice. 

It is very important to have full and complete Drawings before the keel 
of the Vessel is laid, so as to insure unity of action between the Ship and 
Engine Builders, and to afford a clear understanding in reference to con¬ 
tracts. 

The Drawings will contain all new and useful improvements. 

JOHN W. NYSTROM, 

Civil Engineer, 

Letters will be promptly attended to. Philadelphia. 


Mr. Nystrom. the Author of this Pocket-Book, has been in my employ 
since the year 1849, as Constructor and Draughtsman, in which capacity 
he has given me the greatest satisfaction. During that time my atten¬ 
tion was constantly called to the readiness and accuracy of his calculations, 
which were made by means of a Pocket-Book then in manuscript. I have 
frequently requested him to publish it, and am now gratified by receiving 
its pages in print. 


In it there is a Drawing of my Propeller, illustrating the expanding 
pitch first adopted by me, and now generally used. This is the principle 
upon which Propellers have been constructed, under my direction by 
Mr. Nystrom. ^ 

R. F. LOPER. 

March , 1854. 


J 










Contents. 


CONTENTS. 


Algebra, an introduction explaining how to insert numerical values in 
algebraical formulas, page 13. 


A 


PAGE 

C 

PAGE 

Accelerated motiou 

. 

ISO 

Cables and anchors 

- 159 

Acres 

- 

70 

Cables, hemp, strength of 

- 158 

Advertisement 

• 

55 

Calculator, Nystrum’s - 

- 55 

Age, moon’s 

. 

279 

Caloiic. sensible and latent - 224 

Air. composition of 

. 

222 

Calorimeter in steam boilers - 253 

Air-pump - 

• 

236 

Capacity and weigh t of substances 114 

Air-pump valves * 

. 

237 

Cask,4o calculate cubic contents 102 

Alligation- 

. 

20 

Castings, weight of 

- 160 

Altovs ..... 

- 

171 

Catenation ... 

- 143. 146 

Almanac for the 19th century 

275 

Cements ------ 

- ' 172 

Alphabets for printing headings 

280 

Centre of gravity 

- 194 

Angles - 


72 

,, gyration 

- 190 

Animal strength - 


150 

„ percussion - 

- 194 

Annuities 


• 68 

Centifugal force - 

- 186 

Apparent time 


276 

Chains and ropes - 

- 158 

Apothecaries’ weight - 


71 

Chain, surveying - 

- 70 

Area of plane figures - 


96 

Charge of powder 

. 184 

Area of circles 


104 

Circle .... 

- 88 

Arithmetic ... 


11 

Circular saw ... 

• 1 £0 

Arithmetical progression 


64 

Circumference of circles 

. 104 

Astronomy - - - 


274 

Coal, weight and bulk of 

- 257 

Atmosphere - 


214 

Coefficient of vessels - 

- 247 

Attraction - - 


182 

Cog-wheels - - ‘ - 

- 166 

Avoirdupoise weight * 


71 

Cohesive strength 

- 157 

Axles,, strength of 


169 

Coins, gold, silver and copper - 72 




Collision of bodies 

- 151,152 




Colours, water- 

- 69 

^ B 



Combination 

- 20 




Command the engineer 

- 243 

Balls, piling of 

- 

65 

Compass, the mariner’s 

- 206 

„ weight and capacity of 

160 

Compound interest 

. 68 

Balloon - 


217 

Compressive strength- ‘ 

- 156 

Barometer - - 


215 

Concave lenses 

- 260 

Barrel, capacity of 


71 

„ mirrors - 

- 25S 

Beams, greatest strength of 

161 

Condensing Water 

- 236 

Bells, to construct - - 

. 

222 

Condeusor, fresh water 

- 253 


Binomial-theorem ... 22 

Birmingham gauge - - 179, 223 

Blowing off salt water - - * 2+2 

Boiler iron, thickness and weight 256 
Boilers, marine .... 254 
„ weight of - 256 

Boiling point - 227 

Bolts and nuts, proportion of - 180 
Bramah’s hydraulic press • - 203 

Brick - - - - - - 165 

Bulk of substances - - - 198 

Bushel, capacity of - - 70 


Conducting power of heat - - 227 

j Conic sections .... 135 

Constructions, geometrical SO 

Cord of wood .... 105 

Consumption of coal ... 254 
Convex lens - - - 262 

„ mirror ' - - - - 260 

Crank ...... 170 

Cube and cube-root of numbers- 23 
Cubic conteuts of solids - - 100 

Curves for railroads - - 116 

Cut-off steam .... 239 

Cut-off valve.241 

Cycle of the sun. .... 274 

Cycloid.135, 1S5 

Cyma - • - - • - 84 















vf Contents. 


D 


PAGE 



PAGE 

Dav and night, length of 

. 


276 

Governor ... 


- 

187 

Decimals, fractions to vulgar 


120 

Gravitation ... 


- 

182 

Declination of the sun 

- 


278 

Gravity, specific - 


* 

198 

Differential calculs 

• 


11 

,, centre of - 


- 

194 

Diamond - 

• 


72 

Gauge, Birmingham 


- 

179 

Discount ... 

- 


18 

Gyration, centre of 


- 

190 

Displacement of vessels 

- 


249 

H 




Distances on the American coast 

272 




„ between capital cities 


271 

Heat .... 

. 

• 

224 

by railroads in U. S. 


273 

High water, time of - 

• 

• 

28J 

„ of objects at sea 

- 


267 

Heights measured by barometer 

215 

„ spherical 

- 


265 

Helix of a screw - 

• 

• 

94 

Drain, motion of water in 

• 


209 

Hemp, cables strength of 

• 

• 

158 

Dredging machinery - 

- 


154 

Horse power ... 

- 

147, 

149 

Dynamics ... 

- 


147 

Hose, velocity of water through 

207 

E 




Hydraulic press - 

- 

- 

203 

Earth, dimensions of - 
Eccentrics ... 

- 

- 

262 

240 

Hydraulics ... 
Hydrodynamics - 
Hydrometer ... 

- 

- 

206 

210 

202 

Ellips - 
Embankment 

87, 

98, 

118 

Hydrostatics - 

- 

• 

202 

Engineers command - 


- 

243 

I 




Epact ... - 


- 

280 

Incrustation in boilers 

• 

- 

243 

Equation of time 


- 

279 

Inclined plane 

• 

139, 

144 

Evolution ... 


- 

22 

inches and feet - 


- 

120 

Excavation - 


* 

119 

1 ml ix of refraction 

- 

- 

259 

Expansion of air - 


* 

225 

Injection water - » 

- 

- 

237 

„ liquids - 


- 

2: > 5 

integral calculs • » 

w 

• 

11 

„ solids 


- 

221 

Interest, compound 

w 

. 

68 

„ steam 


- 

239 

., simple - 

- 

. 

17 

F 




Involution ... 

- 

• 

21 

Falling bodies 



1S1 

L 




Fathoms ... 



70 

Lateral strength - 

• 

161, 

162 

Feed pump - 



235 

Latitude and longitude 

- 

- 

269 

Fellowship ... 



18 

Law of gravity 

- 

- 

182 

Five engine ... 



207 

Lenses, optical, magnifying power 260 

Fire grate ... 



254 

Letters for printing - 

• 

- 

282 

Fire surface ... 



255 

Lever .... 

• 

139 140 

Flanges, proportion of 



169 

Light .... 

- 

- 

219 

Floating bodies 



204 

Logarithm ... 

• 

• 

56 

Flour mill ... 



150 

„ of numbers 

. 

• 

68 

Flues, weight and thickness of 


171 

„ sin. cos. tang. 

T og line ... 

• 

- 

60 

Fly wheels ... 



193 

• 

. 

70 

Focal distance of lenses 



260 

Longitude into time - 

. 

. 

263 

„ „ mirrors 



257 

Longemetry ... 

- 

• 

98 

Force pump ... 



235 

Lunar cycle - 

• 


274 

Foreign measures and weights 


73 

M 




„ money - 

Fractions, reduction of 



72 

120 

Magnifying power of lenses 


261 

Fresh water condensor 



253 

„ opera-glasses 



261 

Friction ... 



154 

., telescopes 



261 

Fulcrum ... 



139 

Maine value - - 



240 

Funicular machine 



143 

Manoevre the engine - 



243 

G 



Mariners’ compass 



266 




Mathematics 



11 

Gallon, capacity of 


. 

70 

Measures and weights 



70 

Gearing ... 

- 141, 166 

Mechanics ... 



139 

Geography - 



262 

Miles, statute and nautical 


70 

Geometry ... 



78 

Mills, wind - 



218 

Geometrical progression 



66 

Mirrors ... 



258 

Gold metal ... 



171 

Momentum ... 



139 

Golden number * * 



274 

Money .... 



72 



















CONTENTS. 


Tii 


Mean time - 
Moon - 
Moon’s ape - 
„ faces, quarters, etc. 
Mortar „ 

Music ...... 

N 

Natural sin. cos. 

„ tang. cot. 

» toes cosec. 
Navigation 
Night and day 
Nominal horse power 
Nystrom’s calculator 


Opera glass - 
Optics 

Oscillation, centre of 


PAGE 

■ 272 
274 

• 279 
> 278 

• 172 

• 223 


122 

124 

126 

264 

276 

149 

55 


261 

257 

188 


It value of, to 127 decimals - 88 

Paddle wheels .... 248 
Paper, drawing and tracing - 69 

Parabola 87, 138 

Parabolic mirror .... 258 
Pattern makers’ rule ... 168 
Pendulum ..... 188 
Percussion, centre of • 194 

Permutation 19 

Piling of balls and shells - - 65 

Pipe s-teain ..... 238 
Pipes, velocity of water through 207 
Pitch of propellers ... 246 

„ screw .... 93 

„ spiral .... 94 

„ teeth .... 166 

Planimetry - .... 98 

Planetary system ... 277 

Plane, inclined ... 139, 144 

Plane sailing .... 264 
Polygons 79, 103 

Polyhedrons - - - * 95 

Poncelets wheel - - * * 211 

Population in countries and cities 268 
„ on the earth • - 262 

Ports, steam .... 238 
Powder, charge of - * 184 

,, effect of ... 184 

Power, horse ... 147, 149 

Power in moving bodies • * 184 

„ of water .... 210 
Press, hydraulic .... 203 
Printing letters for headings - 280 


Provision « . 

Pulleys . 
Pump, air and feed 


Quantity 

Quadrangle 


Q 


R 


Radiating power of heat 
Railroad curves - 
Range of a cannon ball 
Rebate - - - 

Reduce inches to fret 
Reflecting power of heat 
Reflection of light 
Refraction of light 
Resistance to vessels 
Resultant of forces 
Retarted motion 
Ringing hells - „ 

Roman notations 
Ropes, strength and weight of 


Safety valve - . - 236 

Sailing distances - - 270, 272 

Salt water in boilers ... 242 
San Jacinto .... 250 

Saw mill - - • - - - 150 

Screws, force by - - - - 145 

,, helix .... 94 

„ propeller ... 214 
„ proportion of 180 

Secant natural .... 122 

Segments, table .... Ill 

Shafts, strength of 170 

Shrinkage of castings - - 160 

Signs, algebraical - . - 12 

Simple interest - - - - 17 

Sin. cos. natural .... 122 

Slide valves .... 240 

Slip of propellers ... 247 
Smelting point ... 227 

Solders ..... 171 

Solving triangles mechanically - 130 
Solidity, to calculate - • - 100 

Sound ...... 219 

Sounding ..... 281 

Specific gravity ... 198, 200 
„ caloric ... 228 

Spherical trigonometry • « 131 

„ distances ... 265 

Spiral.87,94 

Square of circles - - • • 94 

Square of numbers * . • 23 

Stability of floating bodies - * 204 


Progression, arithmetical 



64 

Statics 

- 


- 139 

„ geometrical 

o 


66 

Steam 

. 


- 230 

Propellers, screw 

• 


244 

ft 

boilers 


- 254 

Propeller steamers 



150 

ft 

condensor 


- 253 

„ „ of war 



250 

ft 

ports 


- 238 

Proportions 



15 

ft 

table 


. 232 


PAGE 

■ 77 
142 
236 

11 

79 

227 

116 

185 

18 

120 

227 

258 

259 
248 
141 
183 
220 

12 

158 





















CONTENTS. 



PATH 


PAGE 

Steam, formulas - 

- 235 

U. S. steamer-of Avar Wabash - 251 

„ condensor- * 

- 253 

„ standard weight it measure 70 

„ weight of - 

- 235 



Steamers ... 

- 150 

Y 


,, of Avar - 

- 250 

Valves of air pumps - 

- 237 

Steamboat, speed of 

- 249 

,, safety * 

- 238 

Steel, tempering of 

- 171 

„ slide ... 

- 240 

Stereometry 

- 100 

Vein, contracted Avater 

- 209 

Strength of materials 

• • 158 

Vein of water is a parabola 

- 209 

,, „ animals - 

- 150 

Vessels, resistance to - 

- 248 

Stub ends ... 

- 100 

tonnage of 

- 249 

Stuffing box ... 

- 109 

Velocity .... 

- 147 

Sun set and rise - 

- 264 

„ of water 

2C<5, 209 

Surface of solids - 

- 99 

Volume expansion 

- ' 225 

Surface condenser 

- 253 

Vulgar fractions - 

- 120 

Surveying chain - 

- 70 





W 


Jl 


Walking beam ... 

S3, 103 

Talon, to construct a - 

- 84 

Water, composition of - 

219 

Tangent, natural 

- 124 

„ colours ... 

- 09 

Thermometer 

- 220 

„ fresh condenser 

- 253 

Teeth for gearing 

165, 1 CO 

„ power ... 

- 210 

Telescope ... 

- 261 

,, Avheels ... 

211, 213 

Temperature on the earth 

- 220 

Wedge ----- 

- 145 

„ of substances 

- 227 

Weight and measure - 

- 70 

Tempering of steel 

- 171 

„ capacity of baits 

- ICO 

Threshing machine 

- 150 

„ of round and square iron 172 

Tide, pise of - 

- 280 

,, „ wrought il'on pipes 

- 173 

Time, apparent - 

- 276 

„ ,, cast iron pipes 

- 174 

„ mean 

- 279 

„ ,, cast iron cylinders 

- 173 

,, difference in 

* 203 

„ ,, plate rolled iron - 

- 175 

Tonnage of vessels 

- 249 

„ „ cupper bolts 

- 178 

Ton, cwt, pounds 

- 71 

„ pel-square feet 

- 119 

Ton, coal, capacity of - 

- 258 

„ and capacity 

- 114 

Tracing paper . - . - 

- 69 

„ „ bulk 

- 198 

Traveling distances 

271, 273 

„ of castings 

- 160 

Triangles, formulas 

- 123 

steamboilers 

- 254 

„ different kinds 

- 79 

,, „ tubes-, lap-welded 

- 256 

„ by diagrams 

- 130 

Weirs - - - - 

- 210 

Trigonometry, plane. - 

- 120 

Wind ------ 

- 216 

„ spherical 

- 131 

,, mills - - 

- 218 

Threads per inch 

- ISO 

force by 

- 217 

Troy weight , - 

- 71 

Wheels, Avafeer 

211, 213 

Tubes, lap weld, weialit, thickness 

paddle - 

- 248 

and price of 

- 256 

Wood, cord - - 

- 105 

U 


Y 


United States steamer of war San 

Yard, feet, inches - . 

- ' 70 

Jacinto - - - - * - 

I * * 

- 250 

Years, different kind - 

- 274 


• \r 

- . . 1' , 



TtillQj »« 

i * * * w > : 


. - ioea 

•»l>£r09 u 

1 '- saw lo" T 




• • • • 


























COVTKNTS OP Pl,*TFS. 


ix. 

■ ■ ■ . ■ ■ - 


CONTENTS OF PLATES. 


PLATE I. Page 

'Fo reduce inches to decimals of a foot, and fractions of a foot to inches 120 

PLATE IL 

To reduce vulgar fractions to decimals, and decimals to the nearest 

desired vulgar fraction ........ 121 


PLATE III. 

To solve triangles by a Diagram, - • • t 

PLATE IV. 

Construction of teeth for gearing .... 

PLATE V. 


Stub-ends for connecting-rods • 

PLATE VI. 


Stuffing boxes and flanges 


PLATE VII. 

Bells, to construct a ringing bell of any weight and tone 

PLATE VIII. 


Slide-valves fcr Steam Engines 

PLATE IX. 

Eccentrics for slide-valve motion 

PLATE X. 

Screw-Propeller, to construct » 


130 


168 


168 


168 


222 


240 


241 


244 











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Mathematics. 


11 


INTRODUCTION. 

Quantify is that which can be increased or diminished by augments or 
abatements of homogeneous parts. Quantities are of two essential kinds, 
Ge >metrical and Physical. 

1 st, Gmmetrical quantities are those which occupy space; as lines, surfaces, 
solids, liquids , gases, &c. 

2nd, Physical quantities are those which exist in the time but occupy no space, 
they are known by their character and action upon geometrical quantities; as 
attraction , light, heat, electricity and magnetism, colours, force , power, <fcc., &c. 

To obtain the magnitude of a quantity we compare it with a part of the same, 
this part is imprinted in our mind' as a unit, by which the whole is measured 
and conceived. No quantity can be'measured by a quantity of another kind, 
but any quantity can be compared -with any-otlier quantity, and by such com¬ 
parison arises what we call calculation or Mathematics. 

-*4- 

MATHEMATICS. 

Mathematics is a science by which the comparative value of quantities 

are investigated; it is divided into : 

1st, Arithmetic,—that branch of Mathematics, which treats of the nature 
and property of numbers; it is subdivided into Addition, Subtraction, Multiplica¬ 
tion, Division, Involution, Evolution and Logarithms. 

2nd, Algeh»vs»,—that branch of Mathematics which employs letters to repre¬ 
sent quantities, and by that means performs solutions without knowing or 
noticing the value of the quantities. The subdivisions of Algebra are the same 
as in Arithmetic. 

3rd, Geometry,—that branch of Mathematics which investigates the rela¬ 
tive property of quantities that occupies space; its subdivisions are Longernelry, 
Planimetry. Stereometry, Trigonometry, and Conic Sections . 

4tli, Different ial-cale«ls,—that branch of Mathematics, which ascer¬ 
tains the mean effect, produced by group of continued variable causes. 

5t,h, Integral-caiculs,—the contrary of Differential, or that branch of 
Mathematics which investigates the nature of a continued variable cause, that 
has produced a known effect. 

-*-► *--- 

ARITHMETIC. 

The art of manceuvcring numbers, and to investigate the relationship of 
quantities. 

Figures —1, 2, 3. 4, 5, 0, 7, 8, 9. Arabic dignets, about nine hundred years old. 

Cphers —0, D, 0. Sometimes called noughts, it is the beginning of figures and 
things. 

Number is the expression of one or more figures and ciphers. 

Integer is a whole number or unit. 

Fraction is a part of a number or unit. 

When figures .are .joined together in a number, the relative dignity expressed 
by each figure, depends upon its position to the others. Thus, 


W 



674,385; 496,345 ; 695,216; 505,310: 685,3 67; 






12 


Notation. 


Notation is the setting down of any number by figures and ciphers. 
Numeration is the reading of any number in words, from the expression of 
figures and ciphers. 

Characters which describe the operation by numbers ( significations ). 

= Equality, as 6=6, reads 6 is equal to 6. 

+ Plus , Addition, ------ 

— Minus , Subtraction, ------ 

X Multiplication, ------ 

- 5 - or : Division, ------- 


sj Square root, 
\'/ Cube root, 
> Greater, - 
< Less, - 


3+6=9 
• 6—2=4 
3X4=12 
15 : 5=3 
y/ 9=3 
^ 8=2 
- 8 > 4 
6 < 9 




ROMAN NOTATION. 

The Romans expressed their numbers by various repetitions and combin¬ 
ations of seven letters in the alphabet; as, 

1 = 1 . 


2 = 11 . 

3=111. 

4=IY. 

5=V. 

6 =VI. 

7= VII. 

8 =VIII. 

9=IX 
10=X. 

20=XX. 

30=XXX. 

40=XL. 

50= L. 

60= LX. 

70=LXX. 

80=LXXX. 

90=XC. 

100 =C. 

500=D, or LG. 

1 ,000=M, or CO. 

2 ,000=MM, or TTOOO. 

5 .000=V, or LOO- 
6 .000= VI, or MMM. 

10.000=x, or COO. 

50.000=L. or LOGO- 
60.000=LX, or MMMO. 

100.000=5, or COOO. 

1,000,000=M. or COOOO. 

2,000,000 -MM, or MMOOO. 

Examples.— 1854, MDCCCLIV. 

524,365, DXXIVCCCLXV. 

An imperfection in the Roman Notation consists in that, there is no significa¬ 
tion for the cipher as in the Arabic notation 


i As often as a character is repeated, 
so many times is its value repeated. 


{ A less character before a greater 
diminishes its value, as IV=I from 
V, or 1 substracted from 5=4. 


C A less character after a greater in- 
■< creases its value, as XI=X+I, or 
( 10 + 1 = 11 . 


f For every O annexed, this becomes 
(10 times as mauy. 


A bar. thus, — over any number, increases 
‘ it 1000 times. 



















Algebra. 


13 


ALGEBRA. 

In Algebra, we employ certain characters or letters to represent quantities. 
These characters are separated by signs, which describe the operations ; and by 
that means, simplify the solution. 

1. Whatever the value of any quantity may be, it can be represented by a 
character, as a. Another quantity of the same kind, but of different value, be¬ 
ing represented by 6. The sum of these two quantities is of the same kind but 
of different value. 

For Addition we have the algebraical sign +, (plus) which, when placed 
between quantities, denotes they shall be added; as a+6, reads in the 
algebraical language, “a plus 6,” or a is to be added to b. 

Another algebraical sign —, (Equal) denotes that quantities which are placed 
on each side of this sign, are equal. Let the sum of a and b be denoted by the 
letter c; then we have, 

* , 'i ci cH-&=e. 

This composition is called an algebraical equation. The quantity on each side 
of the equal sign is called a member, as a-\-b, is one member, and c, the other. When 
one of the members contains only one quantity, that member is generally 
placed on the first side of the equal sign, and its value commonly unknown; 
but the value of the quantities in the other member being given, as a=4, and 
6 =5, then the practical mode, to insert numerical values in algebraical equa¬ 
tions, will appear; as, 

Equation, c—a+b, 

4+5=9, the value of c. 

2. The sum of three quantities a, b, and c, is equal to d, then 

Equation , d=a+6+c, 

4+5+9=18, the value of d. 

3. For Subtraction we have the algebraical sign,—, (minus) which,'when 
placed before a quantity, denotes it is to be subtracted as, a — b, reads in the 
algebraical language “a minus 6,” or from a, subtract b. Let the difference be 
denoted by the letter c; and a=8. b— 3 

Equation , c=a— b, 

8 —3=5, the value of c. 

4. From the sum of a and b, subtract c, and the result will be d ; then, 

Equation, d=a-\-b — c, 

8+3—5=6, the value of d. 

5. When two equal quantities are to be added, as a+a, it is the same as to 
take one of them twice, and is marked thus 2a. The number 2 is called the 
coefficient of the quantity a. If there are more than two equal quantities to be 
added, the coefficient denotes how many there are of them; as, 

Equation, - - - - a+a=2a, 

** a+a+a=3a, 

“ a+a+a+a=4a, + 

(6c., (6c. 

When the quantities are separated by the signs, plus, or minus, they are 
called terms. 

6 . Multiplication. —When a quantity a, is to be multiplied by another 
quantity b, then a and b are called factors; and separated by no sign as ab; 
which denotes that a is to be multiplied by b; but when the values of a and 6 
are expressed by numbers, they are separated by the sign X (Multiplication); the 
result from Multiplication is called the product. Let a=8, and 6=6, and the pro¬ 
duct of a and b, to be c, then, 

Equation, c=ab, 

8X6=48, the value of c. 

7. The product of a and 6, is to be multiplied by c, and the latter product will 
be equal to d ; then, 

Equation, d=abc, 

8X6X^8=2304, the value of d. 

_ 


2 










It 


Algebra. 


8. The sum of a and 6, is to be multiplied by c, and the product will be d; 
then, 


Equation, d — c(a-\-b), 

48 (8 +6) = 672 the value of d. 


When the sum of two or more quantities is to be multiplied by another quan¬ 
tity, the sum is to be enclosed in parentheses, and denotes itself to be one factor. 
The other factor is to be placed on the outside of the parentheses, as seen in the 
preceding example. 


9. To the product of a and c, add b, and the result will be d; then, 
Equation, d= ac +6, 

8x48 + 6 = 390 the value of d. 


Be particular to distinguish the two Examples 8, and 9. 

10. The sum of a and b, to be multiplied by the sum of a and c; the product 
will be d; then, 

Equation, <7 = (a+6) (a + c), 

(8+6) (8+48) = 784. 

11. The sum of c and b, to be multiplied by the difference of c and a; the re¬ 
sult will be d; then, 

Equation, d = ( c+ b) (c— a), 

(48+6) (48—8) = 2160. 

12. Division. —When a quantity a, is to be separated into b equal parts, the 
numbers of parts or b, is called the divisor , and the value of each part, is called 
the quotient. The sum of the parts or the whole quantity a, is called the dividend: 
a and b, is separated by the sign : (Division); as a: b, reads in the algebraical 
language, “a divided by 6.” Let the quotient be denoted by the letter c; and 
a=18, 6=6, then, 

Equation, c = a: b, 

18 : 6 = 3 the quotient c. 


In Algebra it is found 'more convenient to set up Division as a fraction, then 
it will appear as, 

13. Divide a, by c, and the quotient will be b. Then, 

a 

Equation, b = —> 

18 

— = 6 the quotient 6. 


14. The product of a and b, to be divided by c ; and the product will be d. 
Then, 

ab 


Equation, d =-» 

c 

18X6 


r 36. 


15. The sum of d and b, to be multiplied by c, and the product divided by a; 
then the result will be e. 


Equation, e = 


c (d+b) 
a ’ 
3 (36+6) 
18 


7. 


16. From the product of a and c, subtract 36; divide the remainder by the 
difference of a, and c; the result will be h. 




Proportion. 


15 


Equation, h 


ac —3 b 
a—c 

18X3—3X6 

18—3 


= 2.4. 


An old man said to a smart boy, “ How old are you?” to which he replied.— 
“To seven times my father’s age add yours, divide the sum by double the 
difference of yours and his, and the result will be my age.” 

Letters will denote, 
a = the old man’s age, 
b — the father’s age, 
c — the boy’s age. Then, 


Equation, c ■■ 


7 fc-fa 


2 (a— b ) 


the boy’s age. 


Now for any number of years of the old man and the father, will be a corres¬ 
ponding age of the boy; suppose, 

a — 73 years the age of the old man, 
b = 57 years the father’s age. 

Require the boy’s age. 


c = 


7X57+73 
2 (73—57) 


=14J years. 




PROPORTION. 

. » 

The relative value of two quantities, is obtained by dividing one into the other, 
and the quotient is called the ratio of their relafionship. If the ratio of two 
quantities is equal to the ratio of two other quantities, they are said to be in the 
same proportion; as, 

a: b = c: d, 

reads in the algebraical language “ a is to b as c is to d.” — a, b, c, and d, are call¬ 
ed terms , of which a is the first, b the second, c the third, and d the fourth term. 
The first and fourth are called “ the outer terms” and the second and third, 
“ the inner terms.” The whole is called an “ analogy.” 

A property in the nature of analogies is, that the product of the outer terms 
ad, is equal to the product of the inner be. Suppose a = 4, b — 9, c = 12, 
d — 27. 

4: 9=12 : 27, 
ad=bc, 4X27=9X12. 

If any one of the four quantities are unknown, its value can be calculated 
by the other three; as, 


a = • 

be 

9X12 

- \ 

d 

27 



ad 

4X27 


0 - 

c 

12 

= 


ad 

4X27 

12. 


b 

9 

d = 

be 

9X!2 

= 27. 





















16 Proportions. 


To Alternate a Proportion. 

If a : b = c : d, 

then a : c = b : d, 
and ad - be. 

To Inverse a Proportion. 

If a : b = c : d 7 
then b : a = d : c t 
and be = ad. 

To Multiply a Proportion. 

If a: b = c : d y 

then na : nb = nc : nd , 

, a b c d 

and — : — = — : — 
n n n n 

To Reduce a Proportion. 

If a: b — c : d y 

then na : nb = me : md y 

, abed 
and — — — j _ 

n n m rri 

Compared Proportions. 

If a : b = c : d, 

and c : d = e : / 

then a : b = e : f 

Continued Proportion. 

If a : b = c : d = e : f y 
then af = be = cd y 
and ad = be, cf — de. 

To Compound Proportions. 

If a : d -= c : d, 

and e : f = g : h, 

then ae : df — eg : dh. 

To Compare Proportions » 

If a : b = c : d t 

and e :f= g : h y 

,, a b c d 

then — 

e / g h 

To Combine a Proportion. 

To Combine a Proportion 
Inversely. 

If a: b = c: d, 

then (a \-b):b= ( c+d) : d, 
and d (a+b) = b (c+d). 

If a : b = c : d, 

then a : b = (a+c): (b+d) y 
and a (b+d = b (a+c). 

To Dissolve a Proportion. 

If a: b = c : d, 

then (a—b) : b = ( c—d ) : d, 
and d (a—b) — b (c—d). 

To Dissolve a Proportion 
Inversely. 

If a : b — c : d, 
then a: b = (a— c ): (b— d) y 
and a (b—d) = b (a—c). 

If a: b = d:c, 

then a : b = — : i- 

c d t 

and -7 : \ = c: d. 
a b 

and ae = bd. 

To Find the mean Proportion, 
a : x = x : b 
then x = y/~ab * 

x is the mean Proportion. 

Proportion of Square and Square 
Root. 

If a : b — c : d, 

then a 2 : b 2 = c 2 : d 2 

and y/a : y/b = ^fc : i . 

Propoi tion of any Power or 
Root. 

If a : b = c : d y 

then a“ : b n = c n : d a t 
and Zf~a : lf~b ~= jfcl. \/ r d. 












Simple Interest. 


17 


SIMPLE INTEREST. 

Interest is a profit on money which is lent for a certain time. 

Letters will denote. 

c = the standing capital, or lent money. 
r — interest on the capital c, 
p = per cent, on 100 in the certain time. 

Analog;/, c:r = 100 : p. 

If p is the per cent, on 100, in one year, then t = time in years for the stand¬ 
ing capital c, and the interest r. 

Analogy, cir = 100 : pt. 

From this analogy we obtain the equations, 


Interest, 

Per cent., 
Capital, 
Time in years, 


cpt 

mT’ 

100 r 



100 r 


100 r 


1 , 

2 , 

3, 

4. 


Now for any question in Simple Interest, there is one equation which gives the 
answer. If the time is given in months, weeks, or days, multiply the 100 cor¬ 
respondingly by 12, 52, 365. 

Example 1. What is the interest on $3789.35, for 3 years and five months, at 
G per cent, per annum ? 

t = 3X12+5 = 41 months, from the Equation 1, we have, 


Interest, 


r — 


3789.35X6X41 

12X100 


=776.81 Dollars. 


Example 2. A capital c = $469.78, gave an interest r = 150.72 dollars, in a 
time t = 4 years and 7 months. Require the per centage per annum ? 

t = 4X12+7 = 55 months, from Equation 2, we have, 


Per cent., 




12X100X150.72 
469.78 X 55 


7 per cent. 


Example 3. What capital is required to give an interest r — 345 Dollars in 6 
years, at 5 per cent, per annum ? From the Equation 3, we have, 


Capital, 




Example 4. A.capital c ?= $2365 shall stand until the interest will be r == 550 
Dollars, at p = 6 per cent, per annum. How long must the capital stand ? 

From the Equation 4, we have, 

, 100X550 _ C1T „ 

Time, t = 2?65><6 = 3.876 years. 

12X0.876 = 10.512 months, 4X0.512 == 2.048 weeks, the time t = 3 years, 10 
months, and 2 weeks. 

2* 











Rebate or Discount.—Fellowship. 


18 


REBATE OR DISCOUNT. 

Rebate or Discount is an allowance on money, which is paid before due. 
a = amount of money to be paid in the time t. By agreement the amount is 
paid with a capital c, at the beginning of the time t, but discounted a Rebate r, at 
p per cent, so that the interest on the capital c, at p per cent., should be equal to 


the Rebate r, in the time t. 


a — c+r. 


Debate, 

apt 

jr = —--j • • • 

100 +pt 

* 5, 

Capital, 

100 

100+7? * * * * 

6, 

Per cent., 

100 (a—c) 

■P — ct ’ 

* 7, 

Time, 

^ 100 (a — c) 

cp ’ 

8, 

Amount 

a ex — (100+p<), • 

* 9, 

Amount, 

a = -^-(100+pO, » 

10. 


Now for any question c in Rebate or Discount, there is one equation that will 

give the answer. 

Example 5. A sum of money, a = 78460 dollars is to be paid after 3 years and 
6 months, but by agreement payment is to be made at the present time. What 
will be the Rebate at 7 per cent. 

„ , , 78460X7 X 3.5 „ 

Rebate, r -ioo + 7x3.5 ~ * 15433 - 91 - 


FELLOWSHIP. 

Fellowship* or partnership, is a rule by which companies ascertain each 
fellow’s profit or loss, by their stock. Each fellow’s part in the stock is called 
his share. The sum of shares is called the stock. 

Fellowships are of two kinds, Simple and Double. 

Simple Fellowship, when there is no regard to the time, the shares or 

stock is employed. 

Letters will denote, 

A — share of either one fellow. 
a = profit or loss on the share A. 

S = stock or the sum of the shares. 
s = gain or loss on the stock S. 

Then A: a = S:s. 


Share , 

Profit or loss, 
Stock, 

Gain or loss , 


. aS 

A= —< 

o- 

aS' 

~A* 


12 , 

13, 

14. 









Permutation. 


19 


Example 1. A person had invested A — $116''!5, in a stock S = $64800, 
which gave a gain of s = 138G4. What will be the profit of the person’s 
share? 


Profit, 


11645X13864 

64800 


$2491.45 


Double Fellowship* When the different shares are employed at a dif¬ 
ferent length of time, each share is multiplied by its time employed, and the 
product is the effect of the share. 

Letters will denote, 

t = time for the employed share A. 

T = meantime for the employed stock S. 
e = effect of the share A. 
a = profit of the effect c. 

E — effect of the stock, 
s === gain of the effect E. Then, 


19, 

- 20 , 
21 , 




e: a = 

-- E: S' 


Effect of A, 

aE 

C = T’ 

- - 15, 

Time , 

aE 

As, 

Profit of e, 

i 

II 

e 

- - 16, 

Share, 

aE 

Wl - « 

ts * 

Effect of S, 

E=~, 

a 

- - 17, 

Meantime, 

811 

II 

s-. 

Gain of E, 

II 

01 f ba 

«# 

- - 18, 

Stock, 

„ es 

^ aT ’ 


- 22 . 


Example 2. A canal is to be dug, and requires an effect E = 76850 (men and 
days) to be accomplished; after that it will give a gain s = 12390 Dollars. An 
employer has A = 168 laborers. How many days must those laborers be em¬ 
ployed at the canal, that the employer will obtain a profit a = $5000 ? 


Time , 




5000X76850 

168X12390 


= 184.6 days. 




PERMUTATION. 

Permutation is to arrange a number of things in every possible position. 

It is commonly used in games. 

Example 1. How many different values can be written by the three ciphers 
1, 2, 3. 

1X2X3 = 6. different values, namely, 

123, 132, 213, 231, 312, 321. 

With any three different ciphers can be written six different values. Any 
three things can be placed in 6 different positions. 

Example 2. How many names can he written by the three syllables mo, ta, 
la ? The answer is,—Motala, Molata, Tamola, Talamo, Lamota, Latamo. 

Example 3. How many words can be written by the five syllables, mul, tip, li, 
ca, tion f 

1X2X3X^X5 = 120 words, the answer. 







sa 


Combination.—Alligation. 


-:— . -—"y ; 

n — p 

The accompanying table shows the 

1 = 1 

permutation of different numbers of 

2 = 2 

things up to 14; which will be con- 

3 = 6 

venient in the next coining examples 

4 = 24 

in combination 

6 = 120 


6 = 720 


7 = 5040 


8 = 40320 


9 = 36288C 



10 = 3628800 

11 = 30916800 

12 = 479001600 

13 = 6227020800 

14 = 87178291200. 


-M- 


COMBINATION. 


Coml>iuationis to arrange a less number of things out of a greater, in 

every possible position. It is cominonly used in games. 

Example 1. How mauy different numbers ean be set up by the nine ciphers, 
|, 2, 3, 4, 5, G, 7, 8, 9, and three ciphers in each number? 

« ’ 9X8X7 


1X2X3 


— 84 different numbers. 


Example 2. How many different variations can a player obtain his cards, when 
the set contains 52 cards, of which he receives 8 at a time ? 


52X 51X50X49 X48X47 X46X45 


752538150 variations. 


IX 2X 3X 4X 5X 6X 7X 8 

If they are four players, and pr. 4 = 24, they can play 24X752538160 
=18,060:915,600 different plays. 

If it takes half an hour for each play, and they play 8 hours per'day, it will take 
1S060915600 

-——-— = 1128807225 days = 3;092,622 years. 

2X8 


ALLIGATION. 

Alligation is to mix together a number of different things of different 
price or value, and ascertain the mean value of the mixture ; or from a given 
moan value of a mixture ascertain the proportion and value of each ingre¬ 
dient. 

Let the different things be a, 6, c, and d, &c., their respective price or value 
per unit, z, y, x, and w, &c. 

A — a+b-\-c+d &c., the sum of the things. 

P = mean value or price per unit of A. Then, 


and 


AP — az-\-by -\-cx-\-dw-\- cCc, 
p_ az+by+cx+dw+cte. 


1, 


2 , 


Examp'.e 1. If 3 gallons of wine at $1.37 per gallon, 2 at $2.18, and 5 at $1.75, 
be mixed together, what is a gallon worth of the mixture ? 

A = 3-J-2-1-5 = 10 gallons. 


P = 


3X1.37 +2X2.18+5X3.75 
■ 10 


= $1.72 per gallon. 










Alligation.—Involution. 


21 


Alligation of two ingredients a and 6, with their respective prices or value per 
unit, 2 and g, 2 >P>y. A = a-f 6. 

a:b — ( P—y ) : (z—P) ... 3, 

& (P-y) . A (P-y) 

“=1 7=py 1 “ a “ = —5=5,p ' ' ' - 4 ’ 45 > 

Example 2. A Silver-smith will mix two sorts of silver, one at 54 and one at 
64 cents per ounce. How much must be taken of each sort to make the mixture 
worth 60 cents per ounce. (Formulas.) P= 60. x = 54. y — 64. 

a : 6 = (60—54) : (64—60) = 6 : 4, or, 

4 ounces at 54 cents ; and 6 ounces at 64 cents. 

Alligation of three ingredients, a , b, and c, with their prices or value per unit, 
z, y and x. 

a' : c’ = ( P—x ) : ( z —P) ...... 6, 

a" : b = (P—y) : (z — P) when z^> P> ?/> x, • - 7, 

6 : c" = (P—x) : ( 2 /—P) when z> P> x, . = 8, 

a = a'-fa", c = c'-fc". 

Example 3. A Farmer will mix wheat at 94 cents per bushel, with barley 
at 72 cents, rye at 64 cents per bushel. How much of each sort must be taken 
to make the mixture worth 80 cents per bushel ? 

(Formula 6.) 2 = 94, y = 72, x = 64, and P = 80. 

a : c — (80—64): (94—80) = 16 :14, 
a ': b = (80—72): (94—80) = 8 :14. 

The wheat a = 16+8 = 24 bushels at 94 cents per bushel. 

“ barley 6 = 14 “ “ 72 “ “ 

“ rye c = 14 « “ 64 “ « 

Alligation of four ingredients a, 6, c, and d, respective prices or value per unit; 
z. y, x, and w. 


a: d = (P-w ): ^-^J when z>y>P>x>w - 


b : c = (P—y ): (®-P 
a' \d— (P—w) : (z —P) 1 


a" : 6 = (P-y) : (z-P ) Vwhen «>P>t/>x>w. 


a' 


m 


c = (P—x) : (z —P) ) 


a = a'-fa"-fa'", 
a: d' — (P — w ): (z —P ) 

b :d" = (P—w): (y—P) >when F>y>x>P>u>. 
c : d!" = (P—w) : (x—P) J 
d = d'pd"-Yd">, 


/ 9, 

1 10, 

-rn, 

■s 12, 

(13, 


•{ 


14, 

15, 

16, 


In the same manner, formulae can be set up for any number of ingredients- 


INVOLUTION. 

Involution is to multiply a number into itself a number of times; each 
product is called the power of the number, and the dignity of the power is 
marked by a small figure called exponent, on the right of the number; thus, 

aXa = a 2 the square of a. 
aX«X« = a 3 the cube of a. 
aXaXaXa = a 4 the bisquare of a. 
aXdXaXaX® — a 5 the fifth power of a. 

<£c., <£c., <£c. 

32 = 3X3 = 9. 

2 3 = 2X2X2 = 8. 

4 4 = 4X4X4X4 = 256. 
c£c., die., <£c. 

Binome is a factor or quantity which contains two terms; as (a-f6.) 






22 


Involution'.—Evolution* 


BinomiaI=Thcorcm is the rule which a binome follows, when it is raised 
to any power. 

When a binome is to be multiplied by itself or any other binome, it is set up 
and performed like the'common multiplication by numbers; thus, 


Example 1. 


ajj } 

-tab+b* (| 

a 2 +a6 

«--j- 2 a 6 -j-b- = (a+ 6 ) 3 . 


Example 2. Suppose a +6 = 358746, and a = 358000, b = 746, then, 
(a+ 6)2 = 3587462, 


+a 2 = 128164000000 

- + 2 a 6 =•• 534136000 

- +62 = 556516 

128698692516 = (a+ 6 ) 3 . 

Ex. 3. (a+ 6) 3 = a 3 +3a 2 6+3a6 2 +6 3 . 

Ex. 4. (a+ 6 j 4 = a 4 +4a 3 6+6a 2 6 2 +4a6 3 +6*. 

Ex. 5. (a— 6) 7 = a"—7a 6 6+21a 5 6 2 —35a 4 6 3 +35a 3 6 4 —a 2 6 5 +7a6 7 — 6 h 

Here you will discover the peculiarities of the BinomiabTheorem, which is thus 
expressed in words: 

1st. The exponent of the first term a in the power, is equal to the exponent 
of the binome; and in every successive term, the exponent of a is decreased by 
1 , until the last term of the exponent of a is 0 , and, therefore, disappears, because 
any quantity raised to no power is equal to 1 , thus, a® = 1 and a* = a. 

2 d. hu the "first term of the power, the exponent of 6 is 0 , and therefore 6 will 
first appear in the second term with the exponent 1 , and in every successive 
term the exponent of 6 is increased by l, until in the last term the exponent will 
be eqdal to. the exponent of the binome. 

3d. The coefficient of 'the second term in the power, is equal to the exponent 
of the binome, and the coefficient of any successive term is equal to the product 
of the coefficient and exponent of a in the foregoing term, divided by the num¬ 
ber of terms before the sought coefficient. 

4th. When the second term in the binome is negative, the first term in the 
power will be positive, the second negative, the third positive, the fourth nega¬ 
tive, &c.,.&c. The odd terms are positive, and the even terms are negative. 

5th. The number of terms in the power is one more than the exponent of the 
binome. 


EVOLUTION. 

IS v ol ution is the reverse of Involution, or to find the number that has pro¬ 
duced a given power. In this case the given power is ealled the number, and 
the number which has produced the given power is called the root of the mtm- 
: her. The symbol y ' is generally placed over the number of which the root is 
:to be extracted. The dignity of the root is placed thus ^ of which the figure 3 
is called the index of the root; for the square roots the index 2 is always 
omitted. 

Example 1. 9 == 3 because 3 2 = 9. 

“ 2. 64 — 4 because 4 3 = 64. 

« 3. ^^531441 = 27 “ 27 4 = 531441, 

(tv., die., 

In the accompanying Table are calculated the Squares, Cubes, Square Boots and 
Cu) e Boots of any number up to 1600. By means of this Table, there will be easy 
rules to find the Square Boot and Cube Boot of numbers exceeding 1600. 














Table of Squares, Cubes, Square and Cube Roots'. 


23 


Number. 

Squares. 

Cubes. 

s/llocts. 

& Roots. 

1 

1 

1 

1-0000000 

1-0000000 

2 

4 

8 

1*4142136 

1-2599210 

3 

9 

27 

1-7320508 

1-4422496 

4 

16 

64 

2-0000000 

1-5874011 

5 

25 

125 

2-2360680 

1-7099759 

6 

36 

216 

2-4494897 

1-8171206 

7 

49 

343 

2-6457513 

1-9129312 

8 

64 

512 

2-8284271 

2-0000000 

9 

81 

729 

3-0000000 

2-0800837 

10 

100 

1000 

3-1622777 

2-1544347 

11 

121 

1331 

3-3166248 

2-2239801 

12 

144 

1728 

3-4641016 

2-2894286 

13 

169 

2197 

3-6055513 

2-3513347 

14 

196 

2744 

3-7416574 

2-4101422 

15 

225 

3375 

3-8729833 

2-4662121 

16 

256 

4096 

4-0000000 

2-5198421 

17 

289 

4913 

4-1231056 

2-5712816 

18 

324 

5832 

4-2426407 

2-6207414 

19 

361 

6859 

4-3588989 

2-6684016 

20 

400 

8000 

4-4721360 

2-7144177 

21 

441 

9261 

4-5825757 

2-7589243 

22 

484 

10648 

4-6904158 

2-S020393 

23 

529 

12167 

4-7958315 

2-8438670 

24 

576 

13824 

4-8989795 

2-8844991 

25 

625 

15625 

5-0000000 

2-9240177 

26 

676 

17576 

5-0990195 

2-9624960 

27 

729 

19683 

5-1961524 

3-0000000 

28 

784 

21952 

5-2915026 

3-0365889 

29 

841 

24389 

5-3851648 

3-0723168 

30 

900 

27000 

5-4772256 

3-1072325 

31 

961 

29791 

5-5677644 

3-1413806 

32 

1024 

32768 

5-6568542 

3-1748021 

33 

1089 

35937 

5-7445626 

3-2075343 

34 

1156 

39304 

5-8309519 

3-2396118 

35 

1225 

42875 

5-9160798 

3-2710663 

36 

1296 

46656 

6-0000000 

3-3019272 

37 

1369 

50653 

6-0827625 

3-3322218 

38 

1444 

54872 

6-1644140 

3-3619754 

39 

1521 

59319 

6-2449980 

3-3912114 

40 

1600 

64000 

6-3245553 

3-4199519 

41 

1681 

68921 

6-4031242 

3-4482172 

42 

1764 

74088 

6-4807407 

3-4760266 

43 

1849 

79507 

6*5574385 

3-5033981 

44 

1936 

85184 

6-6332496 

3-5303483 

45 

2025 

91125 

6-7082039 

3-5568933 

46 

2116 

97336 

6-7823300 

3-5830179 

47 

2209 

103823 

6*8556546 

3-6088261 

48 

2304 

110592 

6-9282032 

3-6342411 

49 

2401 

117649 

7-0000000 

3-6593057 

50 

2500 

125000 

7-0710678 

3-6840314 

51 

2601 

132651 

7-1414284 

3-7084298 

52 

2704 

140608 

7-2111026 

3-7325111 


Reciprocals. 

*100000000 
•500000000 
•333333333 
•250000000 
•200000000 
* 16 6 6 6 6 C> 0 7 
•142S57143 
•125000000 
•111111111 
•100000000 
•090909091 
•0S3333333 
•076928077 
•07142S571 
•066666667 
•062500000 
•058823529 
•055555556 
•052631579 
•050000000 
•047619048 
•045454545 
•043478261 
•041666667 
•040000000 
•03S461538 
•037037037 
•035714286 
•034482759 
•033333333 
•032258065 
•031250000 
•030303030 
•029411765 
•028571429 
•02777777S 
•027027027 
•026315789 
•025641026 
•025000000 
•024390244 
•023809524 
•023255814 
•022727273 
•022222222 
•021739130 
•021276600 
•020833333 
•020408163 
•020000000 | 
•019607843 I 
•019230769 . 
















24 


Table op Squares. Cubes. Square and Cube Roots. 


Number. 

Squares. 

Cubes. 

sT Roots. 

\f Roots. 

Reciprocals. 

53 

2809 

148877 

7*2801099 

3*7562858 

*018867925 

54 

2916 

157464 

7*3484692 

3*7797631 

*018518519 

55 

3025 

166375 

7*4161985 

3*8029525 

*018181818 

56 

3136 

175616 

7*4833148 

3*8258624 

*017857143 

57 

3249 

185193 

7*5498344 

3*8485011 

*017543860 

58 

•3364 

195112 

7*6157731 

3*8708766 

•017241379 

59 

•3481 

205379 

7*6811457 

3*8929965 

•016949153 

60 

3600 

216000 

7*7459667 

3*9148676 

•016666667 

61 

3721 

226981 

7*8102497 

3*9304972 

•016393443 

62 

3844 

238328 

7*8740079 

3*9578915 

•016129032 

63 

3969 

250047 

7*9372539 

3*9790571 

*015873016 

64 

4096 

262144 

8*0000000 

4-0000000 

*015625000 

65 

4225 

274625 

8*0622577 

4*0207256 

*015384615 

66 

4356 

287496 

8*1240384 

4*0412401 

*015151515 

67 

4489 

300763 

8*1853528 

4*0615480 

*014925373 

68 

4624 

314432 

8*2462113 

4*0816551 

*014705882 

69 

4761 

328509 

8*3066239 

4*1015661 

*014492754 

70 

■4900 

343000 

8*3666003 

4*1212853 

*014285714 

71 

•5041 

357911 

8*4261498 

4*1408178 

•014084517 

72 

5184 

373248 

8*4852814 

4*1601676 

•013888889 

73 

5329 

389017 

8*5440037 

4*1793390 

*013698630 

74 

5476 

405224 

8*6023253 

4*1983364 

*013513514 

75 

5625 

421875 

8*6602540 

4*2171633 

*013333333 

76 

5776 

438976 

8*7177979 

4*2358236 

*013157895 

77 

5929 

456533 

8*7749644 

4*2543210 

*012987013 

78 

6084 

474552 

8*8317609 

4*2726586 

*012820513 

79 

6241 

493039 

8*8881944 

4*2908404 

•012658228 

80 

6400 

512000 

8*9442719 

4*3088695 

•012500000 

81 

6561 

531441 

9*0000000 

4*3267487 

•012345679 

82 

6724 

551368 

9*0553851 

4*3444815 

*012195122 

83 

6889 

571787 

9*1104336 

4*3620707 

*012048193 

84 

7056 

592704 

9*1651514 

4*3795191 

*011904762 

85 

7225 

614125 

9*2195445 

4*3968296 

*011764706 

86 

7396 

636056 

9*2736185 

4*4140049 

•011627907 

87 

•7569 

658503 

9*3273791 

4*4310476 

•011494253 

88 

7744 

681472 

9*3808315 

4*4470692 

*011363636 

89 

7921 

704969 

9*4339811 

4*4647451 

•011235955 

90 

8100 

729000 

9*4868330 

4*4814047 

•011111111 

91 

8281 

753571 

9*5393920 

4*4979414 

•010989011 

92 

8464 

778688 

9*5916630 

4*5143574 

•010869565 

93 

8649 

804357 

9*6436508 

4*5306549 

•010752688 

94 

8836 

830584 

9*6953597 

4*5468359 

•010638298 

95 

9025 

857374 

9*7467943 

4*5629026 

*010526316 

96 

9216 

884736 

9*7979590 

4*5788570 

•010416667 

97 

9409 

912673 

9*8488578 

4*5947009 

*010309278 

98 

9604 

941192 

9*8994949 

4*6104363 

*010204082 

99 

9801 

970299 

9*9498744 

4*6260650 

*010101010 

100 

10000 

1000000 

10-0000000 

4*6415888 

*010000000 

101 

10201 

1030301 

10*0498756 

4*6570095 

*009900990 

102 

10404 

1061208 

10*0995049 

4*6723287 

*009803922 

103 

10609 

1092727 

10*1488916 

4*6875482 

*009708738 

1Q4 

10816J 

1124864 

10*1980390 

4*7026694 

•009615385 














Table of Squares, Cubes, Square and Cube Roots. 


25 


-1 

N umber. 

Squares. 

Cubes. 

\T Roots. 

4 —■- 

V Roots. 

Reciprocals. 

105 

11025 

1157625 

10-2469508 

4-7176940 

•009523810 

106 

11236 

1191016 

10-2956301 

4-7326235 

*009433962 

107 

11449 

1225043 

10-3440804 

4-7474594 

*009345794 

108 

11664 

1259712 

10-3923048 

4-7622032 

*009259259 

109 

11881 

1295029 

10-4403065 

4-7768502 

•009174312 

110 

12100 

1331000 

10-4880885 

4-7914199 

•009090009 

111 

12321 

1367631 

10-5356538 

4-8058995 

•009009009 

112 

12544 

1404928 

10-5830052 

4-8202845 

•00S92S571 

113 

12769 

1442897 

10-6301458 

4-8345881 

•008S49558 

114 

12996 

1481544 

10-677 0783 

4-8488076 

•008771930 

115 

13225 

1520875 

10-7238053 

4-8629442 

•008695652 

116 

13456 

1560896 

10-7703296 

4-8769990 

•008020690 

117 

13689 

1601613 

10-8166538 

4-8909732 

•008547009 

118 

13924 

1643032 

10-8627805 

4*9048681 

•008474576 

119 

14161 

1685159 

10-9087121 

4-9186847 

•008403361 

120 

14400 

1728000 

10-9544512 

4-9324242 

*008333333 

121 

14641 

1771561 

11-0000000 

4-9460874 

•008264463 

122 

14834 

1815848 

11-0453610 

4-9596757 

*008196721 

123 

15129 

1860867 

11-0905365 

4-9731898 

•008130081 

124 

15376 

1906624 

11-1355287 

4-9866310 

•008064516 

125 

15625 

1953125 

11-1803399 

5-0000000 

•008000000 

126 

15876 

2000376 

11-2249722 

5-0132979 

•007936508 

127 

16129 

2048383 

11-2694277 

5-0265257 

*007874016 

128 

16384 

2097152 

11-3137085 

5-0396842 

•007812500 

129 

16641 

2146689 

11-3578167 

5-0527743 

•007751938 

130 

16900 

2197000 

11-4017543 

5-0657970 

•007692308 

131 

17161 

2248091 

11-4455231 

5-0787531 

•007633588 

132 

17424 

2299968 

11-4891253 

5-0916434 

•007575758 

133 

17689 

2352637 

11-5325626 

5-1044687 

•007518797 

134 

17956 

2406104 

11-5758369 

5-1172299 

•007462687 

135 

18225 

2460375 

11-6189500 

5-1299278 

•007407407 

136 

18496 

2515456 

11-6619038 

5-1425632 

•007352941 

137 

18769 

2571353 

11-7046999 

5*1551367 

•007299270 

138 

19044 

2628072 

11-7473444 

5-1676493 

•007246377 

139 

19321 

2685619 

11-7898261 

5-1801015 

•007194245 

140 

19600 

2744000 

11-8321596 

5-1924941 

•007142857 

141 

19881 

2803221 

11-8743421 

5-2048279 

•007092199 

142 

20164 

2863288 

11-9163753 

5-2171031 

•007042254 

143 

20449 

292*207 

11-9582607 

5-2293215 

•606993007 

144 

20736 

2985984 

12-0000000 

5-2411828 

•006944444 

145 

21025 

304rb6^5 

12-0415946 

5*2535879 

•006896552 

146 

21316 

3112136 

12-0830460 

5-2656374 

•006849315 

147 

21609 

3176523 

12-1243557 

5-2776321 

•006802721 

148 

21901 

3241792 

12-1655251 

5*2895725 

•006756757 

149 

22201 

3307949 

12-2065556 

5*301 1592 

•006711409 

150 

22500 

33750U0 

12-247 I t- Q 7 

5*3132928 

-006666667 

151 

22801 

3442951 

12-2882057 

5-3250740 

•006622517 

152 

23104 

Sol 1008 

12*3288280 

5-3368033 

•006578947 

153 

23409 

3581577 

12-3693169 

5-3484812 

•006535948 

154 

23716 

3652264 

12-4096736 

5*3601084 

•006493506 

155 

24025 

3723875 

12-4498996 

5-3716854 

•OOGiulOlS 

156 

24336 

3796416 

12-4899960 

5-3832126 

•006410256 
























23- Tabus' of Squares, -Crass, Square-and Cure Roots; 


Number. [ 

Squares. 

1 

Cubes. 

y/ lleots. 

\f Hoots. 

Reciprocals. 

1 57 

24649 

3869893 

12*5299641 

5-3940907 

•006369427 

158 

24564 

3944312 

12-569S051 

5-4061202 

•006329114 

169 

25281 

4019679 

12-6095202 

5-4175015 

•006289308 

160 

25600 

4096000 

12-6491106 

5-4288352 

•006250000 

101 ' 

25921 

4173281 

12-68S5775 

5-4401218 

•006211180 

162 

26244 

425152S 

12-7279221 

5-4513618 

•006172840 

163 

26569 

4330747 

12-7671453 

5-4625556 

•006134969 

164 

26896 

4410944 

12-8062485 

5-4737037 

•006097561 

165 

27225 

4492125 

12-8452326 

5-4848066 

*006060606 

166 

27556 

4574296 

12-8840987 

5-49 5 S 647 

•006024096 

167 

27889 

4657463 

12-9228480 

5-5068784 

•005988024 

1GS 

28224 

4741632 

12-9G14814 

5-5178484 

•005952381 

109 

28561 

4S26S09 

13-0000000 

5-52S7748 

•005917160 

170 

28900 

4913000 

13*0384-048 

5-5396583 

•005S82353 

.171 

29241 

5000211 

13-0766968 

5-5504991 

•005847953 

172 

29581 

5088448 

13-1148770 

5-5612978 

•005813953 

173 

29929 

5177717 

13-1529464 

5-5720546 

•005780347 

174 

30276 

5268024 

13-1909060 

5-5S27702 

•005747126 

.175 

30625 

5359375 

13-2287566 

5-5934447 

•005714286 

176 

30976 

5451776 

13-2664992 

5-6040787 

•605681818 

177 

31329 

5545233 

13-3041347 

5-6146724 

•005649718 

178 

31684 

5639752 

13-3416641 

5-62522G3 

•005617978 

179 

32041 

5735339 

13-3790882 

5-6357408 

•005586592 

180 

32400 

5832000 

13-4164079 

5-6462162 

•005555556 

181 

32761 

5929741 

13-4536240 

5-6566528 

•005524862 

132 

33124 

6028568 

13-4907376 

5-6670511 

•005494505 

183 

33489 

■ 6128487 

13-5277493 

5-0774114 

•0054644S1 

184 

33856 

6229504 

13-5646600 

5-6877340 

•005434783 

185 

34225 

6331625 

13-6014705 

5*0980192 

*005405405 

1S6 

34596 

6434856 

13-6381817 

5-7082675 

•005376344 

187 

34969 

6539203 

13-6747943 

5-7184791 

•005347594 

188 

35344 

6644672 

13-7113092 

5-7286543 

•005319149 

189 

35721 

6751269 

13-7477271 

5-7387936 

•005291005 

190 

36100 

6859000 

13-7840488 

5-7488971 

*005263158 

191 

36481 

6967871 

13-8202750 

5-7589652 

*005235602 

192 

36S64 

7077888 

13-8564065 

5-7689982 

•005208333 

193 

37249 

7189517 

13-8924400 

5-7789966 

•005181347 

194 

37 636 

7301384 

13-9283883 

5-7889604 

•005154639 

195 

38025 

7414875 

13-9642400 

5-7988900 

•005128205 

196 

38416 

7529536 

14-0000000 

5-8087857 

■005102041 

197 

3SS09 

7645373 

14-035668S 

5-8186479 

•005076142 

198 

39204 

7762392 

14-0712473 

5-S2S4867 

•005050505 

199 

39601 

7S80599 

14-1067360 

5-8382725 

•005025126 

200 

40000 

SOOOOOO 

14-1421356 

5-8480355 

•005000000 

201 

40401 

8120601 

14-1774469 

5-8577660 

•004975124 

- 202 

40804 

8242408 

14-2126704 

5-8674673 

•004950495 

203 

41209 

8365427 

14-2478068 

5-8771307 

•004926108 

204 

41616 

8489664 

14-2828569 

. 5-8867653 

•004901961 

205 

42025 

8615125 

14-3178211 

5-8963685 

•004878049 

206 

42436 

8741816 

14-3527001 

5-9059406 

•004854369 

.207 

42849 

8869743 

14-3874946 

5-9154817 

•004S30918 

208 

43264 

8998912 

14-4222051 

5-9249921 

•G04807692 















Table of Squares, Cubes, Square axd Cube Roots. 27 


Number. 

Squares 

Cubes. 

\7 Roots. 

Roots. 

Rcc iprcccls. 

209 

43681 

9129329 

14-4568323 

5-9344721 

•0047846S9 

210 

44100 

9261000 

14-4913767 

5-9439220 

*004761905 

211 

44521 

9393931 

14-52.5S390 

5-9533418 

•004739336 

212 

44944 

9528128 

14-5602198 

5-9627320 

•0047160S 1 

213 

45369 

9663597 

14-5945195 

5-9720926 

•004694S36 

214 

45796 

9800344 

14-6287388 

5-9814240 

*004672897 

215 

46225 

9938375 

14-6628783 

5-9907264 

*004651163 

216 

46656 

10077696 

14-6969385 

6-0000000 

•004629630 

217 

47089 

10218313 

14-7309199 

6-0092450 

*004608295 

213 

47524 

10360232 

14-7648231 

6-0184617 

•004587156 

219 

47961 

10503459 

14-7986486 

6-0276502 

•004566210 

220 

48400 

10648000 

14-8323970 

6-0368107 

*004545455 

221 

4SS41 

10793S61 

14-S660687 

6-0459435 

•004524887 

222 

49284 

10941048 

14-8996644 

6-0550489 

•004504505 

223 

49729 

11089567 

14-9331845 

6-0641270 

•0044S4305 

224 

50176 

11239424 

14-9666295 

6-0731779 

•004464286 

225 

50625 

11390625 

15-0000000 

6-0824020 

•004444444 

226 

51076 

11543176 

15-0332964 

6-0991994 

•004424779 

227 

51529 

11697083 

15-0665192 

6-1001702 

•004405286 

228 

51984 

11852352 

15-0996689 

6-1091147 

•004S85965 

229 

52441 

12008989 

15-1327460 

6-1180332 

•004366812 

230 

52900 

12167000 

15-1657509 

6-1269257 

•004347826 

231 

53361 

12326391 

15-1986842 

6-1357924 

•004329004 

232 

53S24 

12487168 

15-2315462 

6-1446337 

•004310345 

233 

54289 

12649337 

15-2643375 

6-1534495 

•004291S45 

234 

54756 

12812904 

15-2970585 

6-1622401 

•004273504 

235 

55225 ‘ 

12977875 

15-3297097 

6-1710058 

•004255319 

236 

55696 

13144256 

15-3622915 

6-1797466 

•004237288 

237 

50169 

13312053 

15-3948043 

6-1884628 

•004219409 

238 

56644 

13481272 

15-4272486 

6-1971544 

•004201681 

239 

57121 

13651919 

15-4596248 

6-2058218 

•004184100 

240 

57000 

13824000 

15-4919334 

6-2144650 

•004166667 

241 

58081 

13997521 

15-5241747 

6-2230843 

•004149378 

242 

58564 

14172488 

15-5563492 

6-2316797 

•004132231 

243 

59049 

14348907 

15-5884573 

6-2402515 

•004115226 

244 

59536 

14526784 

1 5-6204994 

6-2487998 

•00409S361 

245 

60025 

14706125 

15-6524758 

6-2573248 

•0040S1633 

246 

60516 

14SS6936 

15-6843871 

6-2658266 

•004065041 

247 

61009 

15069223 

15-7162336 

6-2743054 

•0040485S3 

248 

61504 

15252992 

15-7480157 

6-2S27613 

•00403225S 

249 

62001 

15438249 

15-7797338 

6-2911946 

•004016064 

250 

62500 

15625000 

15-8113883 

6-2996053 

•004000000 

251 

63001 

15813251 

15-S429795 

6-3079935 

•0039S4064 ' 

252 

63504 

1600300S 

15-8745079 

6-3163596 

•00396S254 

253 

64009 

16194277 

15-9059737 

6-3247035 

•003952569 

254 

64516 

16387064 

15-9373775 

6-3330256 

•003937008 

255 

65025 

16581375 

15-9687194 

6-3413257 

•003921569 

256 

65536 

16777216 

16-0000000 

6-3496042 

•003906250 

257 

66049 

16974593 

16-0312195 

6-3578611 

‘003891051 

258 

66564 

17173512 

16-0623784 

6-3660968 

•003875969 

259 

67081 

17373979 

16-0931769 

6-3743111 

•003861004 

: 200 

67000 

17576000 

16-1245155 

6-3S25043 ' 

•003846154 


















Tabije of Squares. Cubes. Square and Cube Roots. 


Number. 

Squares. 

Cubes. 

V Roots. 

\T Roots. 

- 

Reciprocals. 

261 

68121 

17779581 

16*1554944 

6-3906765 

•003831418 

262 

68644 

17984728 

16-1864141 

6-3988279 

•003816794 

263 

69169 

18191447 

16-2172747 

6-4069585 

•003802281 

264 

69696 

18399744 

16-2480768 

6-4150687 

•003787879 

265 

70225 

18609625 

16-2788206 

6-4231583 

•003773585 

266 

70756 

18821096 

16-3095064 

6-4312276 

•003759398 

267 

71289 

19034163 

16-3401346 

6-4392767 

•003745318 

263 

71824 

19248832 

16*3707055 

6-4473057 

•003731343 

269 

72361 

19405109 

16-4012195 

6-4553148 

•003717472 

270 

72900 

19683000 

16.4316767 

6-4633041 

•003703704 

271 

73441 

19902511 

16-4620776 

6-4712736 

•003690037 

272 

73984 

20123643 

16-4924225 

6-4792236 

•003676471 

273 

74529 

20346417 

16-5227116 

6-4871541 

•003663004 

274 

75076 

20570824 

16-5529454 

6-4950653 

•003649635 

275 

75625 

20796875 

16-5831240 

6-5029572 

•003636364 

276 

76176 

21024576 

16-6132477 

6-5108300 

•003623188 

277 

76729 

21253933 

16-6433170 

6-5186839 

•003610108 

278 

77284 

21484952 

16-6783320 

6-5265189 

•003597122 

279 

77841 

21717639 

16-7032931 

6-5343351 

•003584229 

280 

78400 

21952000 

16-7332005 

6-5421326 

•003571429 

281 

78961 

22188041 

16-7630546 

6-5499116 

•003558719 

282 

79524 

22425768 

16-7928556 

6-5576722 

•003546099 

283 

S0089 

22665187 

16-8226038 

6-5654144 

-003533569 

284 

80656 

22906304 

16-8522995 

6-5731385 

•003522127 

285 

81225 

23149125 

16-8819430 

6-5808443 

•003508772 

286 

81796 

23393656 

16-9115345 

6-5885323 

•003496503 

287 

82369 

23639903 

16-9410743 

6-5962023 

*003484321 

288 

82944 

23887872 

16-9705627 

6-6038545 

•003472222 

289 

83521 

24137569 

17-0000000 

6-6114890 

•003460208 

290 

84100 

24389000 

17-0293864 

6-6191060 

•003448276 

291 

84681 

24642171 

17-0587221 

6-6267054 

•003436426 

292 

85264 

24897088 

17-0880075 

6-6342874 

*003424658 

293 

85849 

25153757 

17-1172428 

6-6418522 

•003412969 

294 

86436 

25412184 

17-1464282 

6-6493998 

•003401361 

295 

87025 

25672375 

17-1755640 

6-6569302 

•003389831 

296 

87616 

25934836 

17-2046505 

6-6644437 

•003378378 

297 

88209 

26198073 

17-2336879 

6-6719403 

•003367003 

298 

88804 

26463592 

17-2626765 

6.6794200 

•003355705 

299 

89401 

26730899 

17-2916165 

6.6868831 

•003344482 

SCO 

90000 

27000CC0 

17-3205081 

6.6943295 

•003333333 

SOI 

90601 

27270901 

17-3493516 

6.7017593 

•003322259 

302 

91204 

27543608 

17-3781472 

6-7091729 

•003311258 

303 

91809 

27818127 

17-4068952 

6-7165700 

•003301330 

304 

92416 

28094464 

17-4355958 

6-7239508 

•003289474 

305 

93025 

28372625 

17-4642492 

6-7313155 

•003278689 

306 

93636 

2SC52616 

17-4928557 

6-7386641 

•003267974 

307 

94249 

28934443 

17-5214155 

6*7459967 

•003257329 

308 

94864 

29218112 

17-5499288 

6-7533134 

•003246753 

309 

95481 

29503609 

17-5783958 

6-7606143 

•003236246 

i 310 

96106 

29791000 

17-6068169 

6-7678995 

•003225806 

* 311 

96721 

3C0S0231 

17-6351921 

6-7751690 

•003215434 

312 

97344 

30371328 

17-6635217 

6-7824229 

•003205128 



















TAfctE of Squares, Cubes, Square and Cube Roots'. 23 


Number. 

Squares. 

Cubes. 

y/ Roots. 

3 /-- 

V -Roots. 

Reciprocals. 

1 

1 

313 

97969 

30664297 

17-6918060 

6-7S96C13 

•003194888 

314 

9S596 

30959144 

17-7200451 

6*7968844 

*003184713 


315 

99225 

31255875 

17*7482393 

6-S040921 

*003174603 


316 

99850 

31554496 

17*7763888 

6-S112S47 

*003164557 


317 

100489 

31855013 

17-8044938 

6-8184620 

.003151574 


318 

101124 

32157432 

17-8325545 

C-S25-6242 

*003144654 


319 

101701 

32401759 

17-8605711 

6*8327714 

•003134796 


320 

102400 

32768000 

17-8885438 

6-8399037 

•003125000 


321 

103041 

33076161 

17-9164729 

6*8470213 

•003115265 


322 

103084 

33386248 

17*9443584 

6*8541240 

•003105590 


323 

104329 

33698267 

17-9722008 

6*8612120 

•003095975 


324 

104976 

34012224 

18-0000000 

6-8682855 

•003086420 


325 

105025 

34328125 

18-0277564 

6-8753433 

•003076923 


326 

106276 

34645976 

18-0554701 

6-8823888 

*003067485 


327 

106929 

34965783 

18-0831413 

6-88941SS 

*003048104 


328 

107584 

35287552 

18-1107703 

6-8964345 

•003048780 


329 

108241 

35611289 

1S-1383571 

6-9034359 

*003039514 


330 

108900 

35937000 

18-1659021 

6-9104232 

•003030303 


331 

109501 

36204091 

18-1934054 

6-9173964 

*003021148 


332 

110224 

36594368 

1S-2208672 

6-9243556 

*003.012048 


333 

110889 

30926037 

1S-2482876 

6-9313088 

•0030030.03 


334 

111550 

37259704 

18-2750669 

6-9382321 

*002994012 


335 

132225 

37595375 

18-3030052 

6-9451496 

•0029S5075 


336 

1.12890 

37933056 

18-3303028 

6-9520533 

•002976190 


337 

1 13569 

38272753 

18-3575598 

6-9589434 

•002967359 


338 

114244 

38014472 

18-3847763 

6-9658198 

•002958580 


339 

114921 

38958219 

18-4119526 

6-9726826 

•002949853 


340 

115600 

39304000 

18-43908S9 

6-9795321 

•002941176 


341 

110281 

39651821 

18-4661853 

6-9863681 

•002932551 


342 

110964 

40001688 

18-4932420 

6-9931906 

•002923977 


343 

117649 

40353607 

1S-5202592 

7-0000000 

•002935452 


344 

118330 

40707584 

1S-5472370 

7-0067962 

•002906977 


345 

119025 

41083625 

1S-5741756 

7-0135791 

•002S98551 


346 

119710 

41421736 

18-6010752 

7-0203490 

•002S90173 


347 

120409 

41781923 

18-6279360 

7-0271058 

•002881844 


3 48 

1211.04 

42144192 

18-6547581 

7-0338497 

•002873503 


349 

121801 

42508549 

18-6815417 

7-0405860 

'002865330 


350 

122500 

42875000 

18-7082869 

7-0472987 

•002857143 


351 

123201 

43243551 

18*7349340 

7-0540041 

•002849003 


352 

123904 

43614208 

18-7616630 

7-0606967 

'002840909 


353 

124009 

43986977 

18*7882942 

7-0673767 

•002S32S61 


354 

125316 

44361864 

18*8148877 

7-0740440 

•002824859 


355 

126025 

44738S75 

18*8414437 

7-0806988 

•002816901 


356 

120730 

45118016 

18*8679623 

7*0873411 

•002S0S9S9 


357 

127449 

45499293 

18*8944435 

7*0939709 

•002801120 


358 

128104 

45S82712 

18*9208879 

7*1005885 

•002793296 


359 

1288S1 

46268279 

18*9472953 

7-1071937 

•002785515 


360 

129600 

46656000 

18*9736660 

7-1137866 

•002777778 


301 

130321 

47045831 

19-0000000 

7-1203674 

•002770083 


362 

131044 

47437928 

19-0262976 

7-1269360 

•002762431 


363 

131769 

47832147 

19-0525589 

7-1334925 

•002754821 


364 

132496 

48228544 

19*0787840 

7*1400370 1 

•002747253 



3* 
































So' Taei.k of Squares, Cubes, Square and Cube Roots. 


Number. 

H- 

Squares. 

Cubes. 

V Roots. 

| \/ Roots. 

| Reciprocals. 

365 

133225 

48627125 

19*1049732 

7-1465695 

•002739726 

366 

133956 

49027896 

19-1311265 

7-1530901 

•002732240 

367 

1346S9 

49430863 

19-1572441 

7-1595988 

•002724796 

368 

135424 

4PS36032 

19-1833261 

7-1660957 

•002717391 

369 

136161 

50243409 

19-2093727 

7-1725809 

•002710027 

370 

136900 

50653000 

19-2353841 

7-1790544 

•002702703 

371 

137641 

51064811 

19-2613603 

7-1855162 

•002695418 

372 

13S3S4 

51478848 

19-2873015 

7*1919663 

•002688172 

373 

139129 

51895117 

19-3132079 

7-1984050 

•002680965 

374 

139876 

52313624 

19-3390796 

7-2048322 

•002673797 

375 

140625 

52734375 

19-3649167 

7-2112479 

•002666667 

376 

141376 

53157376 

19-3907194 

7-2176522 

•002659574 

377 

142129 

53582633 

19-4164878 

7-2240450 

•002652520 

378 

142884 

54010152 

19-4422221 

7-2304268 

•002645503 

379 

143641 

54439939 

19-4679223 

7-2367972 

•002638521 

380 

144400 

54872000 

19-4935887 

7-2431565 

•002631579 

381 

145161 

55306341 

19-5192213 

7-2495045 

•002624672 

382 

145924 

55742968 

19-5448203 

7-2558415 

•002617801 

3S3 

146689 

56181887 

19-5703858 

7-2621675 

•002610966 

384 

147456 

56623104 

19-5959179 

7-2684824 

•002604167 

385 

148225 

57066625 

19-6214169 

7-2747864 

•002597403 

386 

148996 

57512456 

19-6468827 

7-2810794 

•002590674 

387 

149769 

57960603 

19-6723156 

7-2873617 

•002583979 

388 

1505-14 

58411072 

19-6977156 

7-2936330 

•002577320 

389 

151321 

58863869 

19-7230829 

7-2998936 

•002570694 

390 

152100 

59319000 

19-7484177 

7-3061436 

•002564103 

391 

152881 

59776471 

19-7737199 

7-3123828 

•002557545 

392 

153664 

60236288 

19-7989899 

7-3186114 

•002551020 

393 

154449 

6069S457 

19-8242276 

7-3248295 

•002544529 

394 

155236 

61162984 

19-8494332 

7-3310369 

•002538071 

395 

156025 

61629875 

19-8746069 

7-3372339 

•002531646 

396 

156816 

62099136 

19-8997487 

7-3434205 

•002525253 

397 

157609 

62570773 

19-9248588 

7-3495966 

•002518892 

398 

158404 

63044792 

19-9499373 

7-3557624 

•002512563 

399 

159201 

63521199 

19-9749844 

7-3619178 

•002506266 

400 

160000 

64000000 

20-0000000 

7-3680630 

•002500000 

401 

160801 

64481201 

20-0249844 

7-3741979 

•002493766 

402 

161604 

64964808 

20-0499377 

7-3803227 

•002487562 

403 

162409 

65450827 

20-0748599 

7-3864373 

•002481390 

404 

163216 

65939261 

20-0997512 

7-3925418 

•002475248 

405 

164025 

66430125 

20-1246118 

7-3986363 

•002469136 

406 

164836 

66923416 

20-1494417 

7-4047206 

•002463054 

407 

165649 

67419143 

20-1742410 

7-4107950 

•002457002 

408 

166464 

67917312 

20-1990099 

7-4168595 

•002450980 

409 

167281 

68417929 

20-2237484 

7-4229142 

•002444988 

410 

168100 

68921000 

20-24845*67 

7-4289589 

•002439024 

411 

168921 

69426531 

20-2731349 

7-4349938 

•002433090 

412 

169744 

69934528 

20-2977831 

7-4410189 

•002427184 

413 

170569 

70444997 

20-3224014 

7-4470343 

•002421308 

414 

171396 

70957944 

20-3469899 

7-4530399 

•002415459 

415 

172225 

71473375 

20-3715488 

7-4590359 

•002409639 

416 

173056 

71991296 

20-3960781 

7-4650223 

•002406846 






















Table of Squares. Cubes. Square and Cube Root**. .?i 


.«umber. 

Squares. 

Cubes. 

•J Roots. 

\/ Hoots. 

Reciprocals. 

417 

173889 

72511713 

20-4205779 

7-4709991 

•002398082 

418 

174724 

73034632 

20-4450483 

7-4769664 

•002392344 

419 

175561 

73560059 

20-4694895 

7-4829242 

•002386635 

420 

176400 

74088000 

20-4939015 

7-4888724 

•002380952 

421 

177241 

74618461 

20-5182845 

7-4948113 

•002375297 

422 

178084 

75151448 

20-5426386 

7-5007406 

•002369668 

423 

178929 

75686967 

20-5669638 

7-5066607 

•002364066 

424 

179776 

76225024 

20-5912603 

7-5125715 

•002358491 

425 

JL80625 

76765625 

20-6155281 

7-5184730 

•002352941 

426 

181476 

77308776 

20-6397674 

7-5243652 

•002347418 

427 

182329 

77854483 

20-6639783 

7-5302482 

•002341920 

428 

183184 

78402752 

20-6881609 

7-5361221 

•002336449 

429 

184041 

78953589 

20-7123152 

7-5419867 

•002331002 

430 

184900 

79507000 

20-7364414 

7-5478423 

•002325581 

431 

185761 

80062991 

20-7605395 

7-5536888 

•002320186 

432 

186624 

80621568 

20-7846097 

7*5595263 

•002314815 

433 

187489 

81182737 

20-8086520 

7-5653548 

•002309469 

434 

188356 

81746504 

20-8326667 

7-5711743 

•002304147 

435 

189225 

82312875 

20-8566536 

7-5769849 

•002298851 

436 

190096 

82S81856 

20-8806130 

7-5827865 

•002293578 

437 

190969 

83453453 

20-9045450 

7-5885793 

•002288330 

438 

191844 

84027672 

20-9284495 

7-5943633 

•002283105 

439 

192721 

84604519 

20-9523268 

7-6001385 

•002277904 

440 

193600 

85184000 

20-9761770 

7-6059049 

•002272727 

441 

194481 

85766121 

21-0000000 

7-6116626 

•002267574 

442 

195364 

86350888 

21-0237960 

7-6174116 

•002262443 

443 

198249 

86938307 

21-0475652 

7-6231519 

•002257336 

444 

197136 

87528384 

21-0713075 

7-6288837 

•002252252 

445 

198025 

88121125 

21-0950231 

7-6346067 

•002247191 

446 

198916 

88716536 

21-1187121 

7-6403213 

•002242152 

447 

199809 

89314623 

21-1423745 

7-6460272 

•002237136 

448 

200704 

89915392 

21-1660105 

7-6517247 

•002232143 

449 

201601 

90518849 

21-1896201 

7-6574138 

•002227171 

450 

202500 

91125000 

21-2132034 

7-6630943 

*002222222 

451 

203401 

91733851 

21-2367606 

7-6687665 

•002217295 

452, 

204304 

92345408 

21-2602916 

7-6744303 

•0022123S9 

453 

205209 

92959677 

21-2837967 

7-6800857 

*002207506 

454 

206116 

93576664 

21-3072758 

7-6857328 

*002202643 

455 

207025 

94196375 

21-3307290 

7-6913717 

•002197802 

456 

207936 

94818816 

21-3541565 

7-6970023 

•002192982 

457 

208849 

95443993 

21-3775583 

7-7026246 

•002188184 

458 

209764 

96071912 

21-4009346 

7-7082388 

•002183406 

459 

>,10681 

96702579 

21-4242853 

7-7188448 

•002178649 

460 

211600 

97336000 

21-4476106 

7-7194426 

•002173913 

461 

212521 

97972181 

21-4709106 

7-7250325 

•002169197 

462 

213444 

93611128 

21-4941853 

7-7306141 

•002164502 

463 

211369 

99252847 

21-5174348 

7-7361877 

•002159827 

464 

215296 

99897344 

21-5406592 

7-7417532 

•002155172 

465 

216225 

100544625 

21-5638587 

7-7473109 

•002150538 

466 

217156 

101194696 

21-5870331 

7-7528606 

•002145923 

467 

218089 

101847563 

21-6101828 

7-7584023 

•002141328 

468 

219024 

102503232 

21-6333077 

7-7639361 

•002136752 

















32 


Table tf Squares. Cubes. Square and Cube Hoots. 


N u niter. 

Squares. 

Cutes. 

>/ iiccts. 

tools. 

Hecij rccaJs. 

469 

219961 

103161769 

21-6504078 

7-7094620 

•00213211:6 

470 

220900 

1038230CO 

21 -6794834 

7-7749801 

•001121000 

471 

221841 

104487111 

21-7025344 

7*7804904 

•002128142 

472 

222784 

105154048 

21-7255610 

7-7552928 

•002118044 

473 

223729 

106828817 

21-7485032 

7-7914875 

•0021 14105 

474 

224676 

106496424 

21-7715411 

7*7909745 

•002109105 

475 

225625 

107171875 

21-7944947 

7-8024538 

•002105203 

476 

226576 

107850176 

21-8174242 

7-8079254 

•002100840 

477 

227529 

108531333 

21-8403297 

7-8133822 

•002096486 

47S 

228484 

109215352 

21-8032111 

7-8188456 

*0020 92050 

479 

229441 

109902239 

21-8860686 

7-8242942 

•002087083 

4S0 

280400 

110592000 

21-9089023 

7-8297353 

•002083833 

4S1 

231361 

1112S4041 

21-9317122 

7-8351688 

•002079002 

4S2 

232324 

111980168 

21-9544984 

7-8405949 

•002074689 

483 

233289 

112678587 

21-9772610 

7-8460134 

•002070393 

484 

234256 

113379904 

22-0000000 

7-8514244 

•C02GCG116 

485 

235225 

114084125 

22-0227155 

7-8568281 

•002001856 

486 

236196 

114791256 

22-0454077 

7-8022242 

•002057613 

487 

237169 

115501303 

22-0680765 

7-8676130 

•002053388 

4S8 

238144 

116214272 

22-0907220 

7-8729944 

•002049180 

489 

239121 

116930169 

22-1133444 

7-8783684 

•002044990 

490 

240100 

117649000 

22-1359436 

7-8837352 

•002040816 

491 

2410.81 

118370771 

22-1585198 

7-8890940 

•C 020 S G GG 0 

492 

242064 

119095488 

22-1810730 

7-894446S 

•CC2032620 

493 

243049 

119823157 

22-2036033 

7-8997917 

•002028398 

494 

244C36 

120553784 

22*226lf0S 

7-9051294 

•002024291 

495 

245025 

121287375 

22-2485955 

7-9104599 

•002020202 

496 

246016 

122023936 

22-2710575 

7-9157832 

•002016129 

497 

247009 

122763473 

22-2934968 

7-9210994 

•002012072 

498 

248004 

123505992 

22-3159136 

7-9264085 

•C02008C32 

499 

249001 

124251499 

22-3383079 

7-9317104 

•002004008 

500 

250000 

125000000 

22-3606798 

7-9370053 

•0020CC000 

501 

251001 

125751501 

22-3830293 

7-9422931 

•001996008 

502 

252004 

126506008 

22-4053565 

7-9475739 

•001992032 

503 

253009 

127203527 

22-4276615 

7-9528477 

•0019SS072 

504 

254016 

128024064 

22-4499443 

7*9581144 

•001984127 

505 

255025 

128787025 

22-4722051 

7-9633743 

•001980198 

506 

256036 

129554216 

22-494443S 

7-90S6271 

•001976285 

507 

257049 

130323843 

22-5166605 

7-9738731 

•001972387 

50S 

258004 

131096512 

22-5388553 

7-9791122 

•00196S504 

509 

259081 

131872229 

22-5610283 

7-9843444 

•001964637 

510 

260100 

132651000 

22-5831796 

7-9895697 

•C019C0784 

511 

261121 

133432831 

22-6053091 

7-9947S83 

•001956947 

512 

262144 

134217728 

22-6274170 

8-0000000 

•001953125 

513 

263169 

135005697 

22-6495033 

8-0052049 

•001949318 

514 

264196 

135790744 

22-67156S1 

S-0104032 

•001945525 

515 

265225 

1365.90875 

22-6930114 

S-0155946 

•001941748 

516 

266256 

137388096 

22-7156334 

8-0207794 

•001937984 

517 

267289 

1381SS413 

22-7376341 

S-0259574 

•001934236 

518 

268324 

138991832 

22-7596134 

8-0311287 

•001930502 

519 

269361 

139798359 

22-7815715 

S-0362935 

•001926782 

520 

270400 

140608000 

22-8035085 

S-0414515 

•001923077 


J 


















Table of Squares, Cubes, Square and Cube Roots, 


83 


Number. 

Squares. 

Cubes. 

s/ Roots. 

fy' Roots* 

Reciprocals. 

521 

271411 

141420761 

22-8254244 

8-0466030 

•001919386 

522 

272484 

142236648 

22*8473193 

8-0517479 

•001915709 

623 

273529 

143055667 

22*8691933 

8-0568862 

•001912046 

524 

274576 

143877824 

22*8910463 

8-0620180 

•001908397 

525 

275625 

144703125 

22-9128785 

8-0671432 

•001904762 

526 

276676 

145531576 

22-9346899 

8-0722620 

•001901141 

527 

277729 

146363183 

22-9564S06 

8-0773743 

•001897533 

523 

278784 

147197952 

22-9782506 

8-0824800 

•001S93939 

529 

279841 

148035889 

23*0000000 

8-0875794 

•001890359 

530 

280900 

148877001 

23-0217289 

8-0926723 

•001886792 

531 

281961 

149721291 

23-0434372 

8-0977589 

•001883239 

532 

283024 

150568768 

23-0651252 

8-1028390 

•001879699 

533 

284089 

151419437 

23-0867928 

8-1079128 

•001876173 

534 

285156 

152273304 

23-1084400 

8-1129803 

•001872659 

535 

286225 

153130375 

23-1300670 

8-1180414 

•001869159 

536 

287296 

153990656 

23-1516738 

8-1230962 

•001865672 

537 

2S8369 

154854153 

23-1732605 

8-1281447 

•001862197 

538 

289444 

155720872 

23-1948270 

8-1331870 

•01)1858736 

539 

290521 

156590819 

23-2163735 

8-1382230 

•001855288 

540 

291600 

157464000 

23-2379001 

8-1432529 

•001851852 

541 

292681 

J58340421 

23-2594067 

8-1482765 

"001848429 

542 

293764 

159220088 

23-2808935 

8-1532939 

•001845018 

543 

294849 

160103007 

23-3023604 

8-1583051 

•001841621 

644 

295936 

160989184 

23-3238076 

8-1633102 

•001338235 

545 

297025 

161878625 

23-3452351 

8-1683092 

•001834862 

546 

298116 

162771336 

23-3666429 

8-1733020 

•001831502 

547 

299209 

163667323 

23-3880311 

8-1782888 

•001828154 

548 

300304 

164566592 

23-4093998 

8-1832695 

•001S24818 

549 

301401 

165469149 

23-4307490 

8-1882441 

•001821494 

550 

302500 

166375000 

23*4520788 

8-1932127 

•001818182 

551 

303601 

167284151 

23-4733892 

8-1981753 

•001814882 

552 

304704 

168196608 

23-4946802 

8-2031319 

•001811594 

553 

305809 

169112377 

23-5159520 

8-2080825 

•001808318 

554 

306916 

170031464 

23-5372046 

8-2130271 

•001805054 

555 

308025 

170953875 

23-5584380 

8-2179657 

•001801802 

556 

309136 

171879616 

23-5796522 

8-2228985 

•001798561 

557 

310249 

172808693 

23-6008474 

8-2278254 

•001795332 

558 

311364 

173741112 

23-6220236 

8-2327463 

•001792115 

559 

312481 

174676879 

23-6431808 

8-2376614 

•001788909 

560 

313600 

175616000 

23-6643191 

8-2425706 

•001785714 

561 

314721 

176558481 

23-6854386 

8-2474740 

•001782531 

562 

315844 

177504328 

23-7065392 

8-2523715 

•001779359 

563 

316969 

178453547 

23-7276210 

8-2572635 

•001776199 

564 

31S096 

179406144 

23-7486842 

8-2621492 

•001773050 

565 

319225 

180362125 

23-7697286 

8-2670294 

•001769912 

566 

320356 

181321496 

23-7907545 

8-2719039 

•001766784 

567 

321489 

182284263 

23-8117618 

8-2767726 

•001763668 

568 

322624 

183250432 

23-8327506 

8-2816255 

•001760563 

569 

323761 

184220009 

23-S537209 

8-2864928 

•001757469 

570 

324900 

185193000 

23-8746728 

8-2913444 

•001754386 

571 

326041 

186169411 

23-8956063 

8-2961903 

•001751313 

572 

3271S4 

187149-48 

23-9165215 

8-3010304 

•001748252 

•i 












34 Table of Squares, Cures, Square and Cure Roots. 



i 





Number. 

Squares. 

Cubes. 

\J Roots. 

yj Roots. 

Reciprocals. 

573 

328329 

188132517 

23-9374184 

8-3058651 

•C01745201 

574 

329476 

189119224 

23-9582971 

8-3106941 

•001742160 

575 

330625 

190109375 

23-9791576 

8*3155175 

*001739130 

576 

331776 

191102976 

24-0000000 

8-3203353 

*001736111 

577 

332927 

192100033 

24-0208243 

8-3251475 

*001733102 

578 

334084 

193100552 

24-0416306 

S-3299542 

*001730104 

570 

335241 

194104539 

24-0624188 

8-3347553 

•001727116 

580 

336400 

195112000 

24-0831891 

S-3395509 

*001724138 

581 

337561 

196122941 

24-1039416 

8-3443410 

•001721170 

582 

338724 

197137368 

24-1246762 

8-3491256 

*001718213 

583 

339889 

198155287 

24-1453929 

8-3539047 

*001715266 

584 

341056 

199176704 

24-1660919 

8-3586784 

•001712329 

5S5 

34-2225 

200201625 

24-1867732 

8-3634466 

•001709402 

586 

343396 

201230056 

24-2074369 

8-3682095 

*001706485 

587 

344569 

202262003 

24-2280829 

S-372966S 

*001703578 

58S 

345744 

203297472 

24-2487113 

8-3777188 

•001700680 

589 

346921 

204336469 

24-2693222 

8-3824653 

*001697793 

590 

348100 

205379000 

24-2899156 

8-3872065 

•001694915 

591 

319281 

206425071 

24-3104996 

8-3919428 

•001692047 

592 

350464 

207474688 

24-3310501 

S-390C729 

•0016891S9 

593 

351649 

208527857 

24-3515913 

8-4013981 

•001686341 

594 

352836 

2095S45S4 

24-3721152 

8-4061180 

*001683502 

595 

354025 

210644875 

24-3926218 

8-4108326 

•001680672 

596 

355210 

211708736 

24-4131112 

8-4155419 

•001677852 

597 

356409 

212776178 

24-4335834 

8-4202460 

•001675042 

598 

357604 

213847192 

24-4540385 

8-424944 S 

•001672241 

559 

358801 

214921799 

24-4744765 

S-4296383 

•001669449 

600 

360000 

216000000 

24-4948974 

8-43432 C 7 

•001666667 

601 

361201 

217081801 

24-5153013 

8*4390098 

*001663894 

602 

362404 

218167208 

24-5356883 

8-4436877 

*001661130 

603 

308609 

219256227 

24-5560583 

8-4483605 

*001658375 

604 

364816 

220348864 

24-5764115 

8-4530281 

•G01655629 

605 

866026 

221445125 

24-5967478 

8-4576906 

•001652893 

606 

867236 

222545016 

24-6170673 

8-4623479 

*001050165 

607 

868445 

223648543 

24-0378700 

S-4670001 

•001647446 

608 

869664 

224755712 

24-6576560 

8-4716471 

•001644737 

609 

370881 

225866529 

24-6779254 

8-4762892 

•C01642036 

610 

372100 

226981060 

24-6981781 

8-4809261 

•001639344 

611 

373321 

228099131 

24-7184142 

8-4S55579 

•001636661 

612 

374544 

229220928 

24-7386338 

8-4901848 

•001633987 

613 

375769 

230346397 

24-758836S 

8-4948065 

•001631321 

614 

370996 

231475544 

24-7790234 

8-4994233 

•001628664 

615 

378225 

232008375 

24*7991935 

S-5040350 

•001626016 

616 

379456 

233744896 

24-8193473 

S-5086417 

•001623377 

617 

3S0689 

234885113 

24-8394847 

8-5132435 

•001620746 

618 

381924 

236029032 

24-8596058 

8-5178403 

•001618123 

619 

383161 

287176659 

24-8797106 

8*5224331 

•001615509 

6 JO 

384400 

238828000 

24-8997992 

S-52701S9 

•001612903 

621 

385641 

239483061 

24*9198716 

S-5316009 

•001610306 

622 

8868841 

240(141848 

24-9399278 

S-5361780 

•001607717 

623 

388129 

2-; 1804367 

24-J;>39679 

S-5407501 

•001605136 

624 

3S937G 

242970624 1 

24-9799920 

8-5453173 

•C01602564 
















Taels, of Squares, .Cubes, Squab?., ae> Cube Boot?, 35- 



' 


Number. 

Squares. I 

Cubes. 

\f Hoots. 

v'' Boots. 

Reciprocals. 

625 

390625] 

244140625 

25-0000000 

8*54.98797 

•001600000 

626 

391876 | 

245134376 

25-0199920 

8-5544372 

*001597444 

627 

393129 

246491883 

25-0399681 

8-5589899 

*001594896 

62S 

391384 I 

247673152 

25-0599282 

8-5635377 

*001592357 

029 

395611 

248858189 

25-079S724 

8*5680807 

•001589825 

030 

396900 

250047000 

25-099S008 

8*5726189 

•0015S7302 

631 

398101 

251239591 

25-1197134 

8*5771523 

•0015S47S6 

632 

399424 

252435968 

25-1396102 

8*5810809 

•001582278 

633 

400GS9 

253636137 

25-1594913 

8*5862247 

*001579779 

034 

401956 

254840104 

25-1793566 

8*5907238 

*001577287 

635 

403225 

256047875 

25-1992063 

S-5952380 

*001574803 

636 

401496 

257259456 

25-2190404 

8-5997476 

*001572327 

637 

405769 

258474853 

25-23SS589 

S-6042525 

*001569859 

633 

407014 

259694072 

25-2586019 

8-0087526 

•001567398 

639 

408321 

260917119 

25-2784493 

8-6132480 

•001564945 

6 iO 

409600 

262144000 

25-2982213 

8*0177388 

•0015625,00 

Oil 

410SS1 

263374721 

25*3179778 

8-6222248 

•001560062 

612 

412104 

261609288 

25-33771S9 

8-6267063 

•001557032 

613 

413449 

265847707 

25-3574447 

S-6311830 

•001555210 

014 

414736 

267089984 

25-3771551 

8-6356551 

•001552795 

615 

416125 

268336125 

25-3968502 

8*6401226 

*001550388 

010 

417316 

269585136 

25*4165302 

8-6445S55 

*001547988 

647 

418609 

270840023 

25*4361947 

8*6490437 

•001545595 

618 

419904 

272097792 

25-455S441 

8-6534974 

•001543210 

619 

421201 

273359449 

25-4754784 

8*6579465 

•001540832 

050 

422500 

274625000 

25-4950976 

8-6623911 

•001538462 

051 

423801 

275894451 

25-5147013 

S-66CS310 

*001530098 

652 

425104 

277167808 

25-5342907 

8-0712665 

•001533742 

053 

426109 

278445077 

25-5538647 

8-6756974 

•001531394 

054 

427716 

279726264 

25-5734237 

8-6801237 

•001529052 

055 

429025 

281011375 

25-5929678 

8*6845456 

•001526718 

056 

430336 

282300416 

25-6124969 

S-6889630 

•001524390 

657 

431639 

283593393 

25-0320142 

8-6933759 

•001522070 

053 

432964 

284890312 

25-6515107 

8-6977843 

•001519751 

659 

434281 

286191179 

25-0700953 

8*7021882 

•001517451 

600 

435600 

287496000 

25*6904652 

8*7065877 

•001515152 

661 

436921 

288804781 

25,7099203 

S-7109S27 

*001512859 

602 

438244 

290117528 

25*7293007 

8-7153734 

•001510574 

663 

439569 

291434247 

25-7487804 

S-7197596 

*001.508296 

004 

440896 

292754944 

25-7081975 

8*7241414 

*001506024 

665 

41,2225 

294079625 

25*7875939 

8-72S51S7 

•001503759 

060 

443550 

295408290 

25*8009758 

8*7328918 

*001501502 

667 

444S99 

290740963 

25-8203431 

8-7372604 

*001499250 

608 

440224 

298077032 

25*8456060 

8-7416246 

•001497006 

009 

417501 

299418309 

25*8650343 

8*7459846 

•00149476S 

670 

418900 

300703000 

25-88435S2 

8*7503401 

•001492537 

071 

450241 

302111711 

25-9036077 

8-7546913 

•001490313 

072 

451581 

303,404118 

25-9229628 

8-7590383 

•0014SS095 

073 

452929 

304S212J7 

25-9422435 

8-7633809 

•0014S5SS4 

G74 

454276 

306182024 

25-9015100 

8-7677192 

•001483080 

675 

455625 

307540875 

25-9807621 

8-77205.32 

*00] 481481 

070 

456976 

3.03915776 

20-0000000 

8-7763830 

•00J479290 

- 













30 


Table or Squares, Cubes, Square and Cube Roots. 


Number. 

Squares. 

Cubes. 

y/ Roots. 

V Roots. 

Reciprocals. 

677 

458329 

310288733 

26-0192237 

S-7807084 

•001477105 

678 

459684 

311665752 

26-0384331 

1 8-7850296 

•0C1474926 

679 

461041 

313046839 

26-0576284 

8-7893466 

*0014727 54 

680 

462400 

314432000 

26-0768096 

8-7936593 

•001470588 

681 

463761 

315821241 

26-0959767 

8-7979679 

•001468429 

682 

465124 

317214568 

26-1151297 

S-8022721 

•001466276 

683 

466489 

318611987 

26-1342687 

8-8065722 

•001464129 

684 

467856 

320013504 

26-1533937 

S-81086S1 

•001461988 

685 

469225 

.321419125 

26-1725047 

8-8151598 

•C01459S54 

686 

470596 

322828856 

26-1916017 

8-8194474 

•001457726 

6S7 

471969 

324242703 

26-2106848 

8-8237307 

•001455604 

6S8 

473344 

325060672 

26-2297541 

8-8280099 

•001453488 

689 

474721 

3270S2769 

26-2488095 

8-8322850 

•001451379 

690 

476100 

328509000 

26-2678511 

S-83655 59 

•C01449275 

691 

477481 

329939371 

26-2868789 

8-8408227 

•001447178 

692 

478864 

3313738SS 

26-3058929 

8-8450S54 

•001445087 

693 

480249 

332812557 

26-3248932 

8-8493440 

•001443001 

694 

481636 

334255384 

26-3438797 

8-8535985 

•001440922 

695 

483025 

335702375 

26-3628527 

8-8578489 

•001438849 

696 

4S4416 

337153536 

26-3S18119 

8-8620952 

•0014367S2 

697 

485809 

338608873 

26-4007576 

8-8663375 

•001434720 

698 

487204 

340068392 

26-4196896 

8-8705757 

•001432665 

699 

4S8601 

341532099 

26-4386081 

8-8748099 

•001430615 

700 

490000 

343000000 

26-4575131 

8-8790400 

•001428571 

701 

491401 

344472101 

26-4764046 

8-8832661 

•001426534 

702 

492804 

345948408 

26-4952S26 

8-8874882 

•001424501 

703 

494209 

34742S927 

26-5141472 

8-8917063 

•001422475 

704 

495616 

348913664 

26-5329983 

8-8959204 

•001420455 

705 

497025 

350402625 

26-5518361 

8-9001304 

•001418440 

706 

498436 

351895S16 

26-5706605 

8-9043366 

•001416431 

707 

499849 

353393243 

26-5S94716 

8-9085387 

•001414427 

708 

501264 

354894912 

26-6082694 

8-9127369 

•001412429 

709 

5026S1 

356400829 

26-6270539 

8-9169311 

•001410437 

710 

504100 

357911000 

26-6458252 

8-9211214 

•001408451 

711 

505521 

359425431 

26-6645833 

8-9253078 

•001406470 

712 

506944 

36094412S 

26-6833281 

8-9294902 

•001404494 

713 

508369 

362467097 

26-7020598 

8-9336687 

•001402525 

714 

509796 

363994344 

26-7207784 

8-9378433 

•001400560 

715 

511225 

365525875 

26-7394839 

8-9420140 

•001398601 

716 

512656 

367061696 

26-7531763 

8-9461809 

•001396648 

717 

514089 

368601813 

26-776S557 

8-9503438 

•001394700 

718 

515524 

370146232 

26-7955220 

S-9545029 

•001392758 

719 

516961 

371694959 

26-8141754 

8-9586581 

•001390821 

720 

518400 

373248000 

26-8328157 

8-9628095 

•0013SSS89 

721 

519841 

374805361 

26-8514432 

8-9669570 

•001386963 

722 

5212S4 

376367048 

26-8700577 

8-9711007 

•001385042 

723 

522729 

377933067 

26-8S86593 

8-9752406 

•001333126 

724 

524176 

379503424 

26-9072481 

8-9793766 

•001381215 

725 

525625 

381078125 

26-9258240 

8-9835089 

•001279310 

726 

527076 

382657176 

26-9443372 

8-9876373 

•001377410 

727 

528529 

3S4240583 

26-9629375 

8-9917620 

•001375516 

72S 

529984 

385828352 

26-9814751 

8-9958899 

•001373626 












Table of Squares. Cures, Square and Cube Roots. ?7 


' 

Number. 

Squares. 

1 

Cubes. 

y/ Roots. 

V Hoots. 

Reciprocals. 

729 

531441 

387420489 

27-0000000 

9-0000000 

001371742 

730 

532900 

389017000 

27-0185122 

9-0041134 

•001369863 

731 

534361 

390617891 

27-0370117 

9-0082229 

•001367989 

732 

535824 

392223168 

27"0554985 

9-0123288 

•001366120 

733 

537289 

393832837 

27-0739727 

9-0164309 

•001364256 

731 

538756 

395446904 

27-0924344 

9-0205293 

•001362398 

735 

540225 

397065375 

27-1108834 

9-0246239 

•001360544 

730 

541696 

398688256 

27-1293199 

9-0287149 

•001358696 

737 

543169 

400315553 

27-1477149 

9-0328021 

•001356852 

73S 

544644 

401947272 

27-1661554 

9-0368857 

•001355014 

739 

540121 

403583419 

27-1845544 

9-0409655 

•001353180 

740 

547600 

405224000 

27-2029140 

9-0450419 

•001351351 

741 

549801 

406869021 

27-2213152 

9-0491142 

•001349528 

742 

550504 

408518488 

27-2396769 

9-0531831 

•001347709 

743 

552049 

410172407 

27-2580263 

9-0572482 

•001315895 

744 

553530 

411830784 

27-2763634 

9 0613098 

•001344086 

745 

555025 

413493625 

27-2946881 

9-0653677 

•001342282 

746 

556516 

415160936 

27-3130006 

9-0694220 

•001340483 

747 

558009 

416832723 

27-3313007 

9-0734726 

•001338688 

748 

559504 

418508992 

-27-3495887 

9-0775197 

•001336898 

749 

561001 

420189749 

27-3678644 

9-0815631 

•001335113 

750 

562500 

421875000 

27-3861279 

9-0856030 

•001333333 

751 

564001 

423564751 

27-4043792 

9-0S96352 

•001331558 

752 

565504 

425259008 

27-4226184 

9-0936719 

•001329787 

753 

567009 

420957777 

27-4408455 

9-0977010 

•001328021 

754 

508510 

428661064 

27-4590604 

9-1017265 

•001326260 

755 

570025 

430308875 

27-4772633 

9-1057485 

•001324503 

750 

571536 

432081216 

27-4954542 

9-1097669 

•001322751 

757 

573049 

433798093 

27-5136330 

9-1137818 

•001321004 

758 

574564 

435519512 

27-5317998 

9-1177931 

•001349261 

759 

576081 

437245479 

27-5499546 

9-1218010 

•001317523 

700 

577600 

438976000 

27-5680975 

9-1258053 

•001315789 

701 

579121 

440711081 

27-5862284 

9-1298061 

•001314060 

702 

580644 

442450728 

27-6043475 

9-1338034 

•001312336 

703 

582169 

444194947 

27-6224546 

9-1377971 

•001310616 

704 

583096 

445943744 

27-6405499 

9-1417874 

•00130891)1 

765 

585225 

447697125 

27-6586334 

9-1457742 

•001307190 

760 

580756 

449455096 

27-6767050 

9-1497576 

•001305483 

767 

588289 

451217663 

27-6947648 

9-153737 5 

•001303781 

768 

589824 

452984832 

27-7128129 

9-1577139 

•001302083 

769 

591301 

454756609 

27-7308492 

9-1616869 

•001300390 

770 

592900 

456533000 

27-7488739 

9-1656565 

•001298701 

771 

594441 

458314011 

27-7668868 

9-1696225 

•001297017 

772 

595984 

460099648 

27-7848880 

9-1735852 

•001295387 

773 

597529 

461889917 

27-8028775 

9-1775445 

•001293661 

774 

599070 

463684824 

27-8208555 

9-1815003 

•001291990 

775 

000625 

465484375 

27-8388218 

9-1851527 

•001290323 

776 

602176 

467288576 

27-8567766 

9-1S94018 

•001288660 

777 

603729 

409097433 

27-8747197 

9-1933474 

•001287001 

778 

605284 

470910952 

27-8926514 

9-1072897 

•00128534 7 

779 

606841 

472729139 

27-9105715 

9-2012286 

•001283697 

780 

008400 

474552000 

27-9284801 

9-2051641 

•001282051 


A 


























3^, 


Table of Squares, Cubes, Square and Cube Roots. 


Number. 

Squares. 

Cubes. 

\/ Roots. 

) \/ Roots. 

Reciprocals. 

781 

609961 

476379541 

27-9463772 

9-2090962 

•001280410 

782 

611524 

478211768 

27-9642629 

9-2130250 

•001278772 

783 

613089 

480048687 

27-9821372 

9-2169505 

•001277139 

784 

614656 

481890304 

28-0000000 

9-2208726 

•001275510 

785 

616225 

483736625 

28-0178515 

9-2247914 

•001273885 

786 

617796 

485587656 

28-0356915 

9-2287068 

•001272265 

787 

619369 

487443403 

28-0535203 

9-2326189 

•00127064S 

788 

620944 

489303872 

28*0713377 

9-2365277 

•001269036 

789 

622521 

491169069 

28-0891438 

9-2404333 

•001267427 

790 

624100 

493039000 

28-1069386 

9-2443355 

•001265823 

791 

625681 

494913671 

28-1247222 

9-2482344 

•001264223 

792 

627624 

496793088 

28-1424946 

9-2521300 

•001262626 

793 

628849 

498677257 

28-1602557 

9-2560224 

•001261034 

794 

630436 

500566184 

28-1780056 

9-2599114 

•001259446 

795 

632025 

502459875 

28-1957444 

9-2637973 

•001257862 

796 

633616 

504358336 

28-2134720 

9-2676798 

•0012562S1 

797 

635209 

506261573 

28-2311884 

9-2715592 

•001254705 

798 

636804 

508169592 

28-2488938 

9-2754352 

•001253133 

799 

638401 

510082399 

28-2665881 

9-2793081 

•001251364 

800 

640000 

512000000 

28-2842712 

9-2831777 

•001250000 

801 

641601 

513922401 

28-3019434 

9-2870444 

•001248439 

802 

643204 

515849608 

28-3196045 

9-2909072 

•001246883 

803 

644809 

517781627 

28-3372546 

9-2947671 

•001245330 

804 

646416 

519718464 

28-3548938 

9-2986239 

•001243781 

805 

648025 

521660125 

28-3725219 

9-3024775 

•001242236 

■ 806 

649636 

523606616 

28-3901391 

9-3063278 

•001240695 

807 

651249 

525557943 

28-4077454 

9-3101750 

•001239157 

808 

652864 

527514112 

28-4253408 

9-3140190 

•001237624 

809 

654481 

529475129 

28-4429253 

9-3178599 

•001236094 

810 

656100 

531441000 

28-4604989 

9-3216975 

•001234568 

811 

657721 

533411731 

28-4780617 

9-3255320 

•001233046 

812 

659344 

535387328 

28-4956137 

9-3293634 

•001231527 

SI 3 

660969 

537367797 

28-5131549 

9-3331916 

•001230012 

814 

062596 

539353144 

28-5306852 

9-3370167 

•001228501 

815 

664225 

541343375 

28-5482048 

9-3408886, 

•001226994 

816 

665856 

543338496 

28-5657137 

9-3440575 

•001225499 

817 

667489 

545338513 

28-5832119 

9-3484731 

•001223990 

813 

669124 

547343432 

28-6006993 

9-3522857 

•001222494 

819 

670761 

549353259 

28-6181760 

9-3560952 

•001221001 

820 

672400 

551368000 

28-6356421 

9-3599016 

■001219512 

821 

674041 

553387661 

28-6530976 

9-3637049 

•001218027 

822 

675684 

555412248 

28-6705424 

9-3675051 

•001216545 

823 

677329 

557441767 

28-6879716 

9-3713022 

•001215067 

824 

678976 

559476224 

28-7054002 

9-3750963 

•001213592 

825 

680625 

561515625 

28-7228132 

9-3788873 

•001212121 

826 

632276 

563559976 

28-7402157 

9-3826752 

•001210654 

827 

683929 

565609283 

28-7576077 

9-38-64600 

•001209190 

828 

685584 

567663552 

28-7749891 

9-3902419 

•001207729 

829 

687241 

569722789 

28-7923601 

9-3940206 

•001206273 

830 

688900 

571787000 

28-8097206 

9-3977964 

•001204S19 

831 

690561 

573856191 

28-8270706 

9-4015691 

•001203369 

832 

692224 

575930368 

28-8444102 

9-4053387 

•001201923 

1 






















Table of Squares, Cubes, Square and Cube Roots; 39 


N umber. 

Squares. 

Cubes. 

sj Hoots. 

■s/ Roots. 

Reciprocals. 

S33 

693889 

578009537 

28*8617304 

9-4091054 

•001200480 

834 

695556 

580693*04 

28-8790582 

9-4128690 

•001199041 

835 

697225 

582182875 

28-8963666 

9-4166297 

•001197605 

836 

693896 

584277056 

28-9136646 

9-4203873 

*001196172 

837 

700569 

586376253 

28-9309523 

9-4241420 

•001194748 

838 

702244 

588480472 

28-9482297 

9-4278936 

•001198317 

839 

703921 

590589719 

28-9654967 

9-4316423 

•001191895 

840 

705600 

592704000 

28-9827535 

9-4353800 

•001190476 

841 

707281 

594823321 

29-0000000 

9-4391307 

•001189061 

842 

708964 

596947688 

29-0172363 

9-4428704 

•001187648 

843 

710649 

599077107 

29-0344623 

9-4466072 

•001186240 

844 

712336 

601211584 

29-0516781 

9-4503410 

•001184834 

845 

714025 

603351125 

29-0688837 

9-4540719 

•001183432 

846 

715716 

605495736 

29-0860791 

9-4577999 

*001182033 

847 

717409 

607645423 

29-1032644 

9-4615249 

*001180638 

848 

719104 

609800192 

29-1204396 

9-4052470 

•001179245 

849 

720801 

611960049 

29*1376040 

9-4689661 

•001177856 

850 

722500 

614125000 

29-1547595 

9-4726824 

•001176471 

851 

724201 

616295051 

29-1719043 

9-4763957 

•001175088 

852 

725904 

618470208 

29-1890390 

9-4801061 

•001173709 

853 

727609 

620650477 

29-2061637 

9-4838186 

•001172333 

854 

729316 

622835864 

29-2232784 

9-4875182 

•001170960 

855 

731025 

625026375 

29-2403830 

9-4912200 

•001169591 

856 

732736 

627222016 

29-2574777 

9-4949188 

•001168224 

857 

734449 

629422793 

29-2745623 

9-4986147 

•001166861 

858 

736164 

631628712 

29-2916370 

9-5023078 

•001165501 

S59 

737881 

633839779 

29-3087018 

9-5059980 

•001164144 

860 

739600 

636056000 

29-3257566 

9-5096854 

•001162791 

861 

741321 

638277381 

29-3428015 

9-5133699 

•001161440 

862 

743044 

640503928 

29-3598365 

9-5170515 

•001160093 

863 

744769 

642735647 

29-3768616 

9-5207303 

•001158749 

861 

746196 

644972544 

29-3938769 

9-5244063 

•001157407 

865 

748225 

647214625 

29-4108823 

9-5280794 

•001156069 

866 

749956 

649461896 

29-4278779 

9-5317497 

•001154734. 

867 

751689 

651714363 

29-4448637 

9-5354172 

•001153403 

868 

753124 

653972032 

29-4618397 

9-5390818 

•001152074 

869 

755161 

656234909 

29-4788059 

9-5427437 

•001150748 

870 

756900 

658503000 

29-4957624 

9-5464027 

•001 149425 

871 

758841 

660776311 

29-5127091 

9*5500589 

•001148106 

872 

760384 

663054848 

29-5296461 

9-5537123 

•001146789 

873 

762129 

665338617 

29*5465734 

9-5573630 

•001145475 

874 

763876 

667627624 

29-5034910 

9-5610108 

•001144165 

87 5 

765625 

669921875 

29*5803989 

9-5646559 

•001142857 

876 

767376 

672221376 

29-5972972 

9-5682782 

•001141553 

877 

7691.29 

674526133 

29-6141858 

9-5719377 

•001140251 

878 

770884 

676836152 

29-6310648 

9*5755745 

•001138952 

879 

772641 

679151439 

29-6479342 

9-5792085 

•001187656 

880 

774400 

681472000 

29-6647939 

9-5828397 

•001186864 

881 

770161 

683797841 

29-6816442 

9-5864382 

•001185074 

882 

777924 

686128968 

29-6984848 

9-5900937 

*001188787 

883 

779689 

688465387 

29-7153159 

9-5937169 

•001132503 

8S4 

781456 

690807104 

29-7321375 

9-5973373 

•001131222 






















40 Table of Squares, Cubes, Square and Cube Roots. 


Number. 

Square.-. 

Cubes. 

' V HOOtS. 

v/ Roots. 

j 

Reciprocals. 

885 

783225 

693154125 

29*7489496 

9*6009548 

*001129944 

886 

784996 

695506456 

29*7657521 

9*6045696 

*001128668 

887 

786769 

697864103 

29*7825452 

1 9*6081817 

*001127396 

888 

788544 

700227072 

29*7993289 

9*6117911 

*001126126 

889 

790321 

702595369 

29*8161030 

9*6153977 

*001124859 

890 

792100 

704969000 

29*8328678 

9*6190017 

*001123596 

891 

793881 

707347971 

29*8496231 

9*6226030 

*001122334 

892 

795664 

707932288 

29*8663690 

9*6262016 

*001121076 

893 

797449 

712121957 

29*8831056 

9*6297975 

*001119821 

891 

799236 

714516984 

29*8998328 

9*6333907 

*001118568 

S95 

801025 

716917375 

29*9165506 

9*6369812 

*001117818 

896 

802816 

719323136 

29*9332591 

9*6405690 

*001116071 

897 

804609 

721734273 

29*9499583 

9*6441542 

*001114827 

898 

806404 

724150792 

29*9666481 

9*6477367 

*001113586 

899 

808201 

726572699 

29*9833287 

9*6513166 

*001112347 

900 

810000 

729000000 

30*0000000 

9*6548938 

*001111111 

901 

811801 

731432701 

30*0166621 

9*6584684 

*001109878 

902 

813604 

733870808 

30*0333148 

9*6620403 

*001108647 

903 

815109 

736314327 

30*0499584 

9*6656096 

•001107420 

904 

817216 

738763264 

30*0665928 

9*6691762 

•001106195 

905 

819525 

741217625 

30*0832179 

9*6727403 

*001104972 

906 

820836 

743677416 

30*0998339 

9*6763017 

*001103753 

907 

822649 

746142643 

30*1164407 

9*6798604 

*001102536 

908 

824464 

748613312 

20*1330383 

9*6834166 

*001101322 

909 

826281 

751089429 

30*1496269 

9*6869701 

*001100110 

910 

828100 

753571000 

30*1662063 

9*6905211 

*001098901 

911 

829921 

756058031 

30*1827765 

9*6940694 

*001097695 

912 

831744 

75S550S25 

30*1993377 

9*6976151 

*001096491 

913 

833569 

761048497 

30*2158899 

9*7011583 

*001095290 

914 

835396 

763551944 

30*2324329 

9*7046989 

*001094092 

915 

837225 

766060875 

30*2489669 

9*7082369 

*001092896 

916 

839056 

768575296 

30*2654919 

9*7117723 

*001091703 

917 

840889 

771095213 

30*2820079 

9*7153051 

*001090513 

918 

842724 

773620632 

30*2985148 

9*7188354 

•001089325 

919 

844561 

776151559 

30*3150128 

9*7223631 

*001088139 

920 

846400 

778688000 

30*3315018 

9*7258883 

*001080957 

921 

848241 

781229961 

30*3479818 

9.7294109 

*001085776 

922 

850084 

783777448 

30*3644529 

9*7329309 

*001084599 

923 

851929 

786330467 

30*3809151 

9-73644S4 

•001083423 

924 

853776 

788889024 

30*3973683 

9*7399634 

•0010S2251 

925 

855625 

791453125 

30*4138127 

9*7434758 

*001081081 

926 

857476 

794022776 

30*4302481 

9*7469857 

*001079914 

927 

859329 

796597983 

30*4466747 

9*7504930 

*001078749 

928 

861184 

799178752 

30*4630924 

9*7539979 

*001077586 

929 

863041 

801765089 

30*4795013 

9*7575002 

•001076426 

930 

864900 

801357000 

30*4959014 

9*7610001 

•001075269 

931 

866761 

806954491 

30*5122926 

9*7644974 

*001074114 

932 

868624 

809557568 

30-52S6750 

9*7679922 

*001072961 

933 

870489 

812166237 

30*5450487 

9*7714845 

*001071811 

934 

872356 

814780504 

30*5614136 

9*7749743 

*001070664 

935 

874225 

817400375 

30*5777697 

9*7784616 

*001069519 

936 

876096 

820025856 

30*5941171 

9*7829466 

*00106837 6 























41 


Table of Squares. Cubes, Square and Cube Roots. 


Number, j 

Squares 

Cubes. 

\/ Roots. 

\J Roots. 

Reciprocals. 

937 1 

877969 

S22656953 

30-6104557 

9-7854288 

•001067236 

933 

879844 

825293672 

30-6267857 

9-7889087 

•001066098 

939 

881721 

827936019 

30-6431069 

9-7923861 

•001064963 

940 

883600 

830584000 

30-6594194 

9-7958611 

•001063830 

941 

885481 

833237621 

30-6757233 

9-7993330 

•001062699 

942 

887364 

835896SS8 

30-6920185 

9-802S036 

•001061571 

943 

889249 

838561807 

30-7083051 

9-8062711 

•001060445 

944 

891136 

841232384 

30-7245830 

9-8097362 

•001059322 

945 

893025 

843908625 

30-7408523 

9-8131989 

•001058201 

946 

894916 

846590536 

30-7571130 

9-8166591 

•001057082 

947 

896808 

849278123 

30-7733651 

9-8201169 

•001055966 

948 

898704 

851971392 

30-7896086 

9-8235723 

•001054852 

949 

900601 

854670349 

30-8058436 

9-8270252 

•C01053741 

950 

902500 

857375000 

30-8220700 

9-8304757 

•001052632 

951 

904401 

860085351 

30-8382879 

9-8339238 

•001051525 

952 

906304 

862801408 

30-8544972 

9-8373695 

•001050420 

953 

908209 

865523177 

30-8706981 

9-8408127 

•001049318 

954 

910110 

868250664 

30-8868904 

9-8442536 

•001048218 

955 

912025 

870983875 

30-9030743 

9-8476920 

•001047120 

956 

913936 

873722816 

30-9192477 

9-8511280 

•001046025 

957 

915849 

876467493 

30-9354166 

9-8545617 

•001044932 

958 

917764 

879217912 

30-9515751 

9-8579929 

•001043841 

959 

919081 

881974079 

30-9677251 

9-8614218 

•001042753 

960 

921600 

884736000 

30-9838668 

9-8648483 

•001041667 

961 

923521 

887503681 

31-0000000 

9-8682724 

•001040583 

962 

925444 

890277128 

31-0161248 

9-8716941 

•001039501 

963 

927369 

893056347 

31-0322413 

9-8751135 

•001038422 

964 

929296 

895841344 

31-0483494 

9-8785305 

•001037344 

965 

931225 

898632125 

31-0644491 

9-8819451 

•001036269 

906 

933156 

901428696 

31-0805405 

9-8853574 

•001035197 

967 

935089 

904231063 

31-0966236 

9-8887673 

•001034126 

968 

937024 

907039232 

31-1126984 

9-8921749 

•001033058 

969 

938961 

909853209 

31-1287648 

9-8955801 

•001031992 

970 

940900 

912673000 

31-1448230 

9-8989830 

•001030928 

971 

942841 

915498611 

31-1608729 

9-9023835 

•001029866 

972 

944784 

918330048 

31-1769145* 

9-9057817 

•001028807 

973 

946729 

921167317 

31-1929479 

9-9091776 

•001027749 

974 

948676 

924010424 

31-2089731 

9-9125712 

•001026694 

975 

950625 

926859375 

31-2249900 

9-9159624 

•001025641 

976 

952576 

929714176 

31-2409987 

9-9193513 

•001024590 

977 

954529 

932574833 

31-2569992 

9-9227379 

•001023541 

978 

956484 

935441352 

31-2729915 

9-9261222 

•001022495 

979 

958441 

938313739 

31-2889757 

9-9295042 

•001021450 

980 

960400 

941192000 

31-3049517 

9-9328839 

•001020408 

9S1 

962361 

944076141 

31-3209195 

9-9362613 

•001019168 

982 

964324 

946966168 

31-3368792 

9-9396363 

•001018330 

993 

966289 

949862087 

31-3628308 

9-9430092 

•001017294 

984 

968256 

952763904 

31-3687743 

9-9463797 

•001016260 

985 

970225 

955671625 

31-3847097 

9-9497479 

•001015228 

986 

972196 

958385256 

31-4006369 

9-9531138 

•001014199 

987 

974169 

961504803 

31-4165561 

9-9564775 

•001013171 

988 

976144 

964430272 

31-4324673 

9-9598389 

•001012146 


_ 

4 . 




















42 Table of Squares. Cuffs. Square and Cube Coots. 


Juurl er. 

(ji.iire.-. 

Cutes. 

sj iioots. 

\/ Roots. 

7- 

j Reciprocals. 

989 

978121 

967361669 

31-4483704 

9-9631981 

*001011122 

990 

980100 

970299000 

31-4642654 

9-9665549 

•001010101 

991 

982081 

973242271 

31-4801525 

9-9699055 

•001009082 

992 

984064 

976191488 

31-4960315 

9-9732619 

•001008065 

993 

986049 

979146657 

31-5119025 

9-9766120 

•001007049 

994 

988036 

982107784 

31-5277655 

9-9799599 

•001006036 

995 

990025 

985074875 

31-5436206 

9-9833055 

•001005025 

996 

992016 

988047936 

31-5594677 

9-9866488 

•001004016 

997 

994009 

991026973 

31-5753068 

9-9899900 

•001003009 

998 

996004 

994011992 

31-5911380 

9-9933289 

•001002004 

999 

998001 

997002999 

31-6069613 

9-9966656 

•001001001 

1090 

1000000 

1000000000 

31-6227766 

10-0000000 

•001000000 

1001 

1000201 

1003003001 

31-6385840 

10-0033222 

•0009990010 

1002 

1004004 

1006012008 

31-6543866 

10-0066622 

•0009980040 

1003 

1006009 

1009027027 

31-6701752 

10-0099899 

•0009970090 

1004 

1008016 

1012048064 

31-6859590 

10 0133155 

•0009960159 

1005 

1010025 

1015075125 

31-7017349 

10-0166389 

•0009950249 

1006 

1010036 

1018108216 

31-7175030 

10-0199601 

•0009940358 

1007 

1014049 

1021147343 

31-7332633 

10-0232791 

•0009930487 

1008 

1016064 

102419-2512 

31-7490157 

10-0265958 

•0009920635 

1009 

1018081 

1027243729 

31-7647603 

10-0299104 

•0009910803 

1010 

1020100 

1030301000 

31-7804972 

10-0332228 

•0009900990 

1011 

1020121 

1033364331 

31-7962262 

10-0365330 

•0009891197 

1012 

1024144 

1036433728 

31-8119474 

10-0398410 

-0009881423 

1013 

1026169 

1039509197 

31-8276609 

10-0431469 

•0009871668 

1014 

1028196 

1042590744 

31 -8433666 

10-0464506 

•0009861933 

1015 

1030225 

1045678375 

31-8590646 

10-0497521 

•0009852217 

1016 

1032256 

1048772096 

31-8747549 

10-0530514 

•0009842520 

1017 

1034289 

1051871913 

31-8904374 

10-0563485 

•0009832842 

1018 

1036324 

1054977832 

31-9061123 

10-0596435 

•0009823183 

1019 

1038361 

1058089859 

31-9217794 

10-0629364 

•0009813543 

1020 

1040400 

1061208000 

31 -9374388 

10-0662271 

•0009803922 

1021 

1042441 

1064332261 

31-9530906 

10-0695156 

•0009794319 

1022 

1044484 

1067462648 

31-9687347 

10-0728020 

•0009784736 

1023 

1046529 

1070599167 

31-9843712 

10-0760863 

•0009775171 

1024 

1048576 

1073741821 

32-0000000 

10-0793684 

•0009765625 

1025 

1050625 

1076890625 

32-0156212 

10-0826484 

•0009756098 

1026 

1052676 

1080045576 

32-0312348 

10-0859262 

•0009746589 

1027 

1054729 

1083206683 

32-0468407 

10-0892019 

•0009737098 

1028 

1056784 

1086373952 

32-0624391 

10-0924755 

•0009727626 

1029 

1058841 

1089547389 

32-0780298 

10-0957469 

•0009718173 

1030 

1060900 

1092727000 

32-0936131 

10-0990163 

•0009708738 

1031 

1062961 

1095912791 

32-1091SS7 

10-1022835 

•0009699321 

1032 

1065024 

1099104768 

32-1247568 

10-1055487 

•0009689922 

1033 

1067089 

1102302937 

32-1403173 

10-1088117 

•0009080542 

1034 

1069156 

1105507304 

32-1558704 

10-1120726 

•0009671180 

1035 

1071225 

1108717875 

32-1714159 

10-1153314 

•0009661836 

1036 

1073296 

1111934656 

32-1869539 

10-1185882 

•0009652510 

1037 

107536911115157653 

32-2024844 

10-1218428 

•0009643202 

1038 

1077444 

1118386872 

32-2180074 

10-1250953 

•0009633911 

1039 

1079521 

1121622319 

32-2335229 

10-1283457 

•0009624639 

1040 

L _ 

1081600 

1124864000 

32-2490310 

10-1315941 

•0009615385 




















TablI of Squares. Cubes. Square and Cube Roots. 


43 


Number. 

— 

Squares. 

Cubes. 

yj Roots. 

v' Roots. 

Reciprocals. 

1041 

1083681 

1128111921 

32*2645316 

10-1348403 

•0009606148 

1042 

1085764 

1131366088 

32-2800248 

10-1380845 

•0009596929 

1043 

1087849 

1134626507 

32-2955105 

10-1413266 

•0009587738 

1044 

1089936 

1137893184 

32-3109888 

10-1445667 

•0009578544 

1045 

1092025 

1141166125 

32-3264598 

10-1478047 

•0009569378 

1046 

1094116 

1144445336 

32-3419233 

10-1510406 

•0009560229 

1047 

1096209 

1147730823 

32-3573794 

10-1542744 

•0009551098 

104S 

1098304 

1151022592 

32-3728281 

10-1575062 

•0009541985 

1 049 

1100401 

1154320649 

32-3882695 

10-1607359 

•0009532888 

1050 

1102500 

1157625000 

32-4037035 

10-1639636 

•0009523810 i 

1051 

1104601 

1160935651 

32-4191301 

10-1671893 

•0009514748 

1052 

1106704 

1164252608 

32-4345495 

10-1704129 

•0009505703 

1053 

1108809 

1167575877 

32-4499615 

10-1736344 

•0009496676 

1054 

1110916 

1170905464 

32-4653662 

10-1768539 

*9009487666 

1055 

1113125 

1174241375 

32-4807635 

10-1800714 

•0009478673 

1056 

1115136 

1177583616 

32-4961536 

10-1832868 

•0009469697 

1057 

1117249 

1180932193 

32-5115364 

10-1865002 

•0009460738 

1 OSS 

1119364 

1184287112 

32-5269119 

10-1897116 

•0009451796 

1059 

1121481 

1187648379 

32-5422802 

10-1929209 

•0009442871 

1060 

1123600 

1191016000 

32-5576412 

10-1961283 

•0009433962 

1061 

1125721 

1194389981 

32-5729949 

10-1993336 

*0009425071 

1062 

1127844 

1197770328 

32-5883415 

10-2025369 

•0009416196 

1063 

1129969 

1201157047 

32-6035807 

10-2057382 

•0009407338 

1064 

1132096 

1204550144 

32-6190129 

10-2089375 

•0009398496 

1065 

1134225 

1207949625 

32-6343377 

10-2121347 

•0009389671 

1066 

1136356 

1211355496 

32-6496554 

10-2153300 

•0009380863 

1067 

1138489 

1214767763 

32-6649659 

10-2185233 

•0009372071 

1068 

1140624 

1218186432 

32-6S02693 

10-2217146 

•0009363296 

1069 

1142761 

1221611509 

32-6955654 

10-2249039 

•0009354537 

1070 

1144900 

1225043000 

32-7108544 

10-2280912 

*0009345794 

1071 

1147041 

1228480911 

32-7261363 

10-2312766 

•0009337068 

1072 

1149184 

1231925248 

32-7414111 

10-2344599 

•0009328358 

1073 

1151329 

1235376017 

32-7566787 

10-2376413 

*0009319664 

1074 

1153476 

1238833224 

32-7719392 

10-2408207 

•0009310987 

1075 

1155625 

1242296875 

32-7871926 

10-2439981 

•0009302326 

1076 

1157776 

1245766976 

32-8024398 

10-2471735 

‘0009293680 

1077 

1159929 

1249243533 

32-8176782 

10-2503470 

•0009285051 

1078 

1162084 

1252726552 

32-8329103 

10-2535186 

•0009276438 

1079 

1164241 

1256216039 

32-8481354 

10-2566881 

•0009267841 

1080 

1166400 

1259712000 

32-8633535 

10-2598557 

•0009259259 

1081 

1168561 

1263214441 

32-8785644 

10-2630213 

•0009250694 

1082 

1170724 

1266723368 

32-8937684 

10-2661850 

•0009242144 

1083 

1172889 

1270238787 

32-9089653 

10-2693467 

•0009233610 

1084 

1175056 

1273760704 

32-9241553 

10-2725065 

•0009225092 

1085 

1177225 

1277289125 

32-9393382 

10-2756644 

•0009216590 

1086 

1179396 

1280824056 

32-9545141 

10-2788203 

*0009208103 

1087 

1181569 

1284365503 

32-9696830 

10-2819743 

•0009199632 

1088 

1183744 

1287913472 

32-9848450 

10-2851264 

•0009191176 

1089 

1185921 

1291467969 

33-0000000 

10-2882765 

*0009182736 

1090 

1188100 

1295029000 

33-0151480 

10-2914247 

•0009174312 

1091 

1190281 

1298596571 

33-0302891 

10-2945709 

•0009165903 

1092 

1192464 

1302170688 

33-0454233 

10-2977153 

•0009157509 

















44 


Table of Squares, Cubes, Square axd Cube Roots. 


Number. 

Squares, 

Cubes. 

V Roots. 

Roots. 

Reciprocals. 

1093 

1194649 

1305751357 

33'0605505 

10-300S577 

*0009149131 

1094 

1196S36 

1309338584 

33'075670S 

10*3039982 

*0009140768 

1095 

1199025 

1312932375 

33*0907842 

10*3071368 

*0009132420 

1096 

1201216 

1316532736 

33*1058907 

10*3102735 

*0009124008 

1097 

1203409 

1320139673 

33*1209903 

10*3134083 

*0009115770 

109S 

1205504 

1323753192 

33*1360830 

10*3165411 

*0009107468 

1099 

1207801 

1327373299 

33*1511689 

10*3196721 

*0009099181 

1100 

1210000 

1331000000 

33*1662479 

10-322S012 

*0009090909 

1101 

1212201 

1334633301 

33*1813200 

10*3259284 

*0009082652 

1102 

1214404 

1338273208 

33*1963853 

10*3290537 

*0009074410 

1103 

1216609 

1341919727 

33*2114438 

10*3321770 

*0009066183 

1104 

1218816 

1345572864 

33*2266955 

10*3352985 

*0009057971 

1105 

1221025 

1349232625 

33*2415403 

10*3384181 

*0009049774 

1106 

1223236 

1352899016 

33*25657S3 

10*3415358 

*0009041591 

1107 

1225449 

1356572043 

33*2716095 

10*3446517 

*0009033424 

UOS 

1227664 

1360251712 

33*2866339 

10*3477657 

*0009025271 

1109 

1229881 

1363938029 

33*3016516 

10*3508778 

*0009017133 

1110 

1232100 

1367631000 

33*3166625 

10-35398S0 

*0009009009 

mi 

1234321 

1371330631 

33*3316666 

10*3570964 

*0009000900 

1112 

1236544 

1375036928 

33*3466640 

10-3602029 

*0008992806 

1113 

1238769 

1378749897 

33*3616546 

10*3633076 

*0008984726 

1114 

1240996 

i382469544 

33*3766385 

10-3664103 

*0008976661 

1115 

1243225 

1386195875 

33*3916157 

10*3695113 

*0008968610 

1116 

1245456 

1389928896 

33*4065862 

10-3726103 

*0008960753 

1117 

1247689 

1393668613 

33*4215499 

10*3757076 

•000S952551 

1118 

1249924 

1397415032 

33*4365070 

10*3788030 

*0008944544 

1119 

1252161 

1401168159 

33*4514573 

10*3818965 

*0008936550 

1120 

1254400 

1404928000 

33*4664011 

10-3,849882 

*0008928571 

1121 

1256641 

140S694561 

33*4813381 

10*3880781 

*0008960607 

1122 

1258884 

1412467848 

33*4962684 

10-3911661 

*0008912656 

1123 

1261129 

1416247867 

33*5111921 

10-3942527 

*0008904720 

1124 

1263376 

1420034624 

33*5261092 

10-3973366 

*0008896797 

1125 

1265625 

1423828125 

33*5410196 

10*4004192 

*0008888889 

1126 

1267876 

1427628376 

33*5559234 

10*4034999 

•0008880995 

1127 

1270129 

1431435383 

33-570S206 

10*4065787 

*0008873114 

1128 

1272384 

1435249152 

33*5857112 

10-4096557 

*0008865248 

1129 

1274641 

1439069689 

33*6005952 

10*4127310 

*0008857396 

1130 

1276900 

1442897000 

33*6154726 

10*4158044 

*0008849558 

1131 

1279161 

1446731091 

33*6303434 

10*4188760 

*0008841733 

1132 

1 281 424 

1450571968 

33*6452077 

10*4219458 

*0008833922 

1133 

1283689 

1454419637 

33*6600653 

10-4250138 

*0008826125 

1134 

1285956 

1458274104 

33*6749165 

10-4280800 

•000SS18342 

1135 

1288225 

1462135375 

33*6897610 

10*4311443 

*0008810573 

1136 

1290496 

1466003456 

83*7045991 

10*4342069 

*0008802817 

1137 

1292769 

1469878353 

33*7174306 

10*4372677 

*0008795075 

1138 

1295044 

1473760072 

33*7340556 

10*4403677 

*0008787346 

1139 

1297321 

1477648619 

33*7490741 

10-4433839 

*0008779631 

1140 

1299600 

1481544000 

33-763S860 

10*4464393 

*0008771930 

1141 

1301881 

1485446221 

33*7786915 

10*4494929 

•000S764242 

1142 

130 4164 

1489355288 

33*7934905 

10*4525448 

*0008756567 

1143 

1306449 

1493271207 

33*8082830 

10*4555948 

•000S74S906 

1144 

1308736 

1497193984 

33*8230691 ' 

10*4586431 

•000S741259 























Table of Squares. Cubes, Square and Cube Roots. 


45 


Number. 

Squares. 

— 

Cubes. 

n/ nOOtS. 

4 / Roots. 

| 1145 

1311025 

1501123625 

33-8378486 

10-4616896 

1146 

1313316 

1505060136 

33*85262 IS 

10-4647343 

1147 

1315609 

1509003523 

3 3 *8 G 78884 

10-4677773 

1148 

1317904 

1512953792 

33*8821487 

10-4708158 

1149 

1320201 

1516910949 

33*8969025 

10-4738579 

1150 

1322500 

1520875000 

33-9116499 

10-4768955 

1151 

1324801 

1524845951 

33-9263909 

10-4799314 

1152 

1327104 

1528823808 

33-9411255 

10*4829656 

1153 

1329409 

1532808577 

33*9558537 

10-4859980 

1154 

1331716 

1536800264 

33-9705755 

10-4890280 

1155 

1334025 

1540798875 

33-9852910 

10-4920575 

1156 

1336336 

1544804416 

34-0000000 

10-4950847 

1157 

1338649 

1548816893 

34*0147027 

10-4981101 

1158 

1340964 

1552836312 

34-0293990 

10-5011337 

1159 

1343281 

1556862679 

34-0440890 

10-5041556 

1160 

1345600 

1560896000 

34*0587727 

10-5071757 

1161 

1347921 

1564936281 

34-0734501 

10-5101942 

1162 

1350244 

1568983528 

34-0881211 

10*5132109 

1163 

1352569 

1573037749 

34-0127858 

10-5162259 

1164 

1354896 

1577098944 

34*1174442 

10-5192391 

1165 

1357225 

1581167125 

34-1320963 

10-5222506 

1166 

1359556 

1585242296 

34* 1467422 

10-5252604 

1167 

1361889 

3 589324463 

34*1613817 

10-5282685 

1168 

1364224- 

1593413632 

34*1760150 

10-5312749 

11 69 

1366561 

1597509809 

34*1906420 

10-5342795 

1170 

1368900 

1601613000 

34*2052627 

10-5372825 

1171 

1371241 

1605723211 

34*2198773 

10-5402837 

1172 

1373584 

1609840448 

34*2344855 

10-5432832 

1173 

1375929 

1613964717 

34*2 490875 

10-5462S10 

1174 

1378276 

1618096024 

34*2636834 

10-5492771 

1175 

1380625 

1622234375 

34*2782730 

10-5522715 

1176 

1382976 

1626379776 

34-2928564 

10-5552642 

1177 

1385329 

1630532233 

34-3074336 

10-5582552 

1178 

1387684 

1634691752 

34*3220046 

10-5612445 

1179 

1390041 

1638858339 

34-3365694 

10-5642322 

1180 

1392400 

1643032000 

34-3511281 

10-5672181 

1181 

1394761 

1647212741 

34‘3656S05 

10-5702024 

1182 

1397124 

1651400568 

34*3802268 

10-5731849 

1183 

1399489 

1655595487 

34*3947670 

10-5761658 

1184 

1401856 

1659797504 

34-4093011 

10-5791449 

1185 

1404225 

1664006625 

34*4238289 

10-5821225 

1186 

1406596 

1668222856 

34*4383507 

10-5850983 

1187 

1408969 

1672446203 

34-4528663 

10-5880725 

1188 

1411344 

1676676672 

34*4673759 

10-5910450 

1189 

1413721 

1680914629 

34-481'8793 

10-5940158 

1190 

1416100 

1685159000 

34-4963766 

10-5969850 

1191 

1418431 

1689410871 

34-5108678 

10-5999525 

1192 

1420864 

1693669888 

34-5253530 

10-6029184 

1193 

1423249 

1697936057 

34-5398321 

10-6058826 

1194 

1425636 

17022093S4 

34-5543051 

10-6088451 

1195 

1428025 

1706489875 

34-5687720 

10-6118060 

1196 

1430416 

1710777536 

34-5832329 

10-6147652 


Reciprocals. 

"0008733G2 !■ 
•0008726003 
•0008718396 
•0008710801 
•0008703220 
•0008695652 
•0008688097 
•00086S055G 
•000S673027 
•0008665511 
•0008658009 
•0008650519 
•0008643042 
•0008635579 
•0008628128 
•0008620690 
•0008613244 
•0008605852 
•0008598452 
•0008591065 
•0008583691 
•0008576329 
•0008568980 
•0008561644 
•0008554320 
•0008547009 
•0008539710 
•0008532423 
•0008525149 
•0008517888 
•0008510638 
•0008503401 
•0008496177 
•0008488961 
•0008481764 
•0008471576 
•0008467401 
•0008460237 
•0008453085 
•0008445946 
•0008438819 
•0008431703 
•0008424600 
•0008417508 
•0008410429 
•0008403361 
•0008396306 
•0008389262 
•0008382320 
•0008375209 
•0008368201 
•0008361204 























46 Table of Squares, Cubes, Square and Cube Roots. 


— 

Number 

j Squares. 

Cubes. 

yf ROOIS. 

T-- *- 

\/ Roots. 

Reciprocals. 

1197 

1432809 

1715072373 

34-5976879 

10-6177228 

•0008354219 

1198 

1435204 

1719374392 

34-6121366 

10-6206788 

•0008347245 

1199 

1437601 

1723683599 

34-6265794 

10-6236331 

•000S340284 

1200 

1440000 

1728000000 

34-6410162 

10-6265857 

•000S333333 

1201 

1442401 

1732323601 

34-6554469 

10-6295367 

*0008326395 

1202 

1444804 

1736654408 

34-6698716 

10-6324860 

*0008319468 

1203 

1447209 

1740992427 

34-6842904 

10-6354338 

*0008312552 

120 4 

1449616 

1745337664 

34-6987031 

10-6383799 

*0008305648 

1205 

1452025 

1749690125 

34-7131099 

10-6413244 

•0008298755 

1206 

1454436 

1754049816 

34-7275107 

10-6442672 

•0008291874 

1207 

1456849 

1758416743 

34*7419055 

10-6472085 

•0008285004 

1208 

1459264 

1762790912 

34-7562944 

10-6501480 

•0008278146 

1209 

1461681 

1767172329 

34-7706773 

10-6530860 

•0008271299 

1210 

1464100 

1771561000 

34-7850543 

10-6560223 

•0008264463 

1211 

1466521 

1775956931 

34-7994253 

10-6589570 

•0008257038 

1212 

1468944 

1780360128 

34-8137904 

10-6618902 

•0008250825 

1213 

1471369 

1784770597 

34*8281495 

10-6648217 

•0008244023 

1214 

1473796 

1789188344 

34-8425028 

10-6677516 

•0008237232 

1215 

1476225 

1793613375 

34-8568501 

10-6706799 

•0008230453 

1216 

147S650 

1798045696 

34*8711915 

10-6736060 

•0008223684 

1217 

1481089 

1802485313 

34-8855271 

10-6765317 

•0008216927 

121S 

1483524 

1806932232 

34*8998567 

10-0794552 

•0008210181 

1219 

1485961 

1811386459 

34-9141805 

10-6823771 

•0008203445 

1220 

1488400 

1815848000 

34-0284984 

10-6852073 

•0008196721 

1221 

1490841 

1820316861 

34-9428104 

10-6882160 

■0008190008 

1222 

1493284 

1824793048 

34-9571166 

10-6911331 

•0008183306 

1223 

1495729 

1829276567 

34-9714169 

10-6940486 

•0008176615 

1224 

1498176 

1833764247 

34-9857114 

10-6969625 

•0008169935 

1225 

1500625 

1838265625 

35-0000000 

10-6998748 

•0008163265 

1226 

1 503276 

1842771176 

35-0142828 

10-7027855 

•0008156607 

1227 

1505529 

1847284083 

35-0285598 

10-7056947 

•0008149959 

1228 

1507984 

1851804352 

35*0428309 

10-7086023 

•0008143322 

1229 

1510441 

1856331989 

35-0570963 

10-7115083 

•0008136696 

1230 

1512900 

1860867000 

35-0713558 

10-7144127 

-000S130081 

1231 

1515361 

1865409391 

35*0856096 

10-7173155 

•0008123477 

1232 

1517824 

1869959168 

35*0998575 

10-7202168 

*0008116883 

1233 

1520289 

IS74516337 

35-1140997 

10-7231165 

*0008110300 

1234 

1522756 

1879080904 

35-1283361 

10-7260146 

•0008103728 

1235 

1525225 

1883652875 

35-1425568 

10-7289112 

•0008097166 

1236 

1527696 

1888232256 

35*1567917 

10-7318062 

•0008090615 

1237 

1530169 

1892819053 

35-1710108 

10-7346997 

•0008084074 

1238 

1532644 

1897413272 

35-1852242 

10-7375916 

•0008077544 

1239 

1535121 

1902014919 

35-1994318 

10-7404S19 

•0008071025 

1210 

1537600 

1906624000 

35-2! 36337 

10-7433707 

"0008064516 

1211 

1540081 

1911240521 

35-2278299 

10-7462579 

•0008058018 

1242 

1542564 

1915864488 

35*2420204 

10-7491436 

•0008051530 

1243 

1545049 ' 

1920495907 

35-2562051 

10-7520277 

•0008045052 

121-4 

1547536 

1925134784 

35-2703842 

10-7549103 

•000803S585 

1245 

1550025 1929781125 

35*2845575 

10-7577913 

•0008032129 

1246 

1552521 

1934434936 

35*2987252 

10-7606708 

•0008025682 

1247 

1555009 

1939096223 

35-3128872 

10-^635488 

•0008019246 

1248 

1557504 

1943764992 

35 3270435 

10-7664252 

•0008012821 






























Table of Squares, Cubes, Square and Cube Roots 


47 


Number. 

Squares. 

Cubes. 

\/ Roots. 

Roots. 

Reciprocals. 

1249 

1560001 

1948441249 

35-3411941 

10*7693001 

*0008006405 

1250 

1562500 

1953125000 

35-3553391 

10-7721735 

*0008000000 

1251 

1565001 

1957816251 

35*3694784 

10-7750453 

*0007993605 

1252 

1567504 

1962515008 

35-3836120 

10-7779156 

*0007987220 

1253 

1570009 

1967221277 

35-3977400 

10-7807843 

*0007980846 

1254 

1572516 

1971935064 

35-4118624 

10-7836516 

*0007974482 

1255 

1575025 

1976650375 

35-4259792 

10-7865173 

*0007968127 

1256 

1577536 

1981385216 

35-4400903 

10-7893815 

*0007961783 

1257 

1580049 

1986121593 

35-4541958 

10-7922441 

*0007955449 

1258 

15S2504 

1990865512 

35-4682957 

10-7951053 

*0007949126 

1259 

1585081 

1995616979 

35-4823900 

10-7979649 

*0007942812 

1260 

1587600 

2000376000 

35-4964787 

10-8008230 

*0007936508 

1261 

1590121 

2005142581 

35-5105618 

10-8036797 

*0007930214 

1262 

1592644 

2009916728 

35*5246393 

10-8065348 

*0007923930 

1263 

1595166 

2014698447 

35*5387113 

10-8093884 

*0007917656 

1264 

1597696 

2019487744 

35-5527777 

10-8122404 

*0007911392 

1265 

1600225 

2024284625 

35*5668385 

10-8150909 

•0007905-138 

1266 

1602756 

2029089096 

35*5808937 

10-8179400 

*0007898894 

1267 

1605289 

2033901163 

35*5949434 

10-8207876 

•0007892660 

1268 

1607824 

2038720832 

35-6089876 

10-8236336 

•0007886435 

1269 

1610361 

2043548109 

35-6230262 

10-8264782 

•0007880221 

1270 

1612900 

204S383000 

35-6370593 

10-8293213 

•0007874016 

1271 

1615441 

2053225511 

35-6510869 

10-8321629 

•0007867821 

1272 

1617984 

205S075648 

35-6651090 

10-8350030 

•0007861635 

1273 

1620529 

2062933417 

35-6791255 

10-8378416 

•0007S55460 

1274 

1623076 

2067798824 

35-6931366 

10-8106788 

•0007849294 

1275 

1625625 

2072671875 

3-5*7071421 

10-8435144 

•0007843J37 

1276 

1628176 

2077552576 

35*7211422 

10-8463485 

•0007836991 

1277 

1630729 

2082440933 

35*7351367 

.10-8491812 

•0007830854 

127S 

1633284 

2087336952 

35*7491258 

10-8520125 

•0007824726 

1279 

1635841 

2092240639 

35-7631095 

10-8548422 

•0007818608 

1280 

1638400 

2097152000 

35-7770876 

10-8576704 

•0007812500 

12S1 

1640961 

2102071841 

35-7910603 

10-8604972 

•0007806401 

1282 

1643524 

2106997768 

35*8050276 

10-8633225 

•0007800312 

1283 

1646089 

2111932187 

35-8189894 

10-8661454 

•0007794232 

1284 

1618656 

2116874304 

35*8329457 

10-8689687 

•0007788162 

1285 

1651225 

2121824125 

35*8468966 

10-8717897 

•0007782101 

1286 

1653796 

2126781656 

35*8608421 

10-8746091 

•0007776050 

1287 

1656369 

2131746903 

35-8747822 

10-8774271 

•0007770008 

1288 

1658944 

2136719872 

35-8887169 

10-8802436 

•0007763975 

1289 

1661521 

2141700569 

35-9026461 

10-8830587 

•0007757952 

1290 

1664100 

2146689000 

35-9165699 

10-8858723 

•0007751938 

1291 

16666S1 

21516S5171 

35-9304884 

10-8886845 

•0007745933 

1292 

1669264 

2156689088 

35-9444015 

10-8914952 

•0007739938 

1293 

1671849 

2161700757 

35-9583092 

10-8943044 

*0007733952 

1294 

1674436 

2166720184 

35*9722115 

10-8971123 

•0007727975 

1295 

1677025 

2171747375 

35*9861084 

10-8999186 

•0007722008 

1296 

1679616 

2176782336 

36-0000000 

10-9027235 

•0007716049 

1297 

1682209 

2181825073 

36*0138362 

10-9055269 

•0007710100 

1298 

1684804 

2186875592 

36-0277671 

10-9083290 

•0007704160 

1299 

168740! 

2191933899 

36*0416126 

10*9111296 

•0007698229 

1300 

1690000 

2197000000 

36-0555128 

10*9139287 

•0007692308 



















48 Table of Scares. Cores, Square ani> Cube Roots. 


N umber. 

Squares. 

Cubes. 

Roots. 

| .y Root,.-. 

! Reciprocals. 

1301 

1692601 

2202073901 

36-0693776 

i 10-9167565 

•0007686395 

1302 

1695204 

2207155608 

36-0832371 

10-9195228 

•0007680492 

1303 

1697809 

2212245127 

36-0970913 

i 0-92231 77 

•0007074579 

1304 

1700416 

2217342464 

36-1109402 

10-9251111 

•0007668712 

1305 

1703025 

2222447625 

36-1247837 

10-9279031 

•0007662835 

1306 

1705636 

2227560616 

36-1386220 

10-9300937 

•0007656968 

1307 

1708249 

2232681443 

36-1524550 

10-9334829 

•0007651109 

1308 

17I0S64 

2237810112 

36-1662826 

10-9362706 

•0007645260 j 

1309 

17134S1 

2242946629 

36-1801050 

10-9390569 

•0007639419 

1310 

1716100 

2248091000 

36-1939221 

4 0-9418418 

•0007633588 

1311 

171S721 

2253243231 

36-2077340 

10-914 6253 

*0007627765 

1312 

1721344 

2258403328 

36-2215406 

10-9475074 

•0007621951 

1313 

1723969 

2263571297 

36-2353419 

4 0-9501880 

•0007(51644 6 j 

1314 

1726596 

2268747144 

36-2491379 

10-9529673 

•0007610350 

1 315 

1729225 

2273930875 

36-2626287 

10-9557451 

•0007604563 | 

1316 

1731856 

2279122496 

36-2767143 

10-9585215 

*0007598784 

1317 

1734489 

2284322013 

36-2904246 

4 0-9612965 

•0007593011 

1318 

1737124 

2289529432 

36-3042697 

10-961070! 

■0007587253 

1319 

1739761 

2294744759 

36-3180396 

10*9663423 

•000758150I 

1320 

1742400 

2299968000 

36-3318042 

10-9696131 

•0007575758 

1321 

1745041 

2305199161 

36*3 455637 

10-9723825 

•0007570028 

1322 

1747684 

2310438248 

36-3593179 

10-9751505 

•0007561297 

1323 

1750329 

2315685267 

36*3730670 

10-9770171 

•0007558579 

1324 

1752976 

2320940224 

36-2868108 

10-9806823 

•0007552870 

1325 

17 55625 

2326203125 

36-1005494 

10-9831462 

•0007547170 

1326 

1758276 

2331473976 

36-4 142829 

10 1)862086 

•000754!178 

1327 

1760929 

23367527S3 

36-42801 12 

10*9889696 

•0007535795 

1328 

17 635SI 

2342039552 

36*1117343 

10-9917293 

•0007530120 

1329 

1766241 

2347334289 

36-4551523 

10-99 14876 

•0007524454 

1330 

1768900 

2352637000 

36-4691650 

10-99724 45 

•0007518797 

1331 

1771561 

2357947691 

36-4828727 

1 1-0000000 

•00075 13 MS 

1332 

177422! 

2363266368 

36-4965752 

1 1-0027541 

•0007507508 

1333 

1776889 

2368593037 

36-5102725 

1 i -0055069 

-0007501875 

1334 

1779556 

2373927704 

36-5239647 

1 i -0082583 

•0007496252 

1335 

1782225 

2379270375 

36"5376518 

11-01 10082 

•0007 190037 

1336 

1784896 

2384621056 

36-5512388 

1 1*0137569 

-0007485030 

1337 

1787569 

2389979753 

36"5650! 06 

1 1:0165011 

•00074 79432 

1338 

1790244 

2395346472 

36-5786823 

11-0192500 

•000747384 2 

1339 

1792921 

2400721219 

36-5923 4 89 

11-0219945 

•0007168260 

1340 

1795600 

2406104000 

36-6060104 

11-0247377 

•00074 626.87 

134 L 

1798281 

2411494S21 

86-6196668 

1 1-0274795 

•0007457122 

1342 

1800961 

2416893688 

36-6333181 

1 1 -0302!99 

•0007451565 

1343 

1803649 

2422300607 

36-6469144 

1 1-0329590 

•0007 146011! 

1344 

1806336 

2427715584 

36-6606056 

1 1 -0356967 

■00074 10476 

1345 

1S09025 

2433138625 

36-6742416 

1 1-0384330 

•0007434941 

1346 

1811716 

2438569736 

36-6878726 

1 1-0411680 

•0007429421 

1347 

1814409 

2444008923 

36-7014986 

1 1-04390! 7 

•0007423905 

1348 

1817104 

2449456192 

36-7151 195 

1 1-0466339 

-0007418398 

1349 

1819801 

2454911549 

36*7287353 

11-0193619 

•0007412898 

1350 

1822500 

2460375000 

36-7423461 

1 1-052094 5 

•0097407407 

1351 

1825201 

2465846551 

36-7559519 

1 1-0548227 

-0007401921 

1352 

L 

1827904 

2471326208 

36-7695526 

11-0575:197 

•0007396450 

1 


































49 


Table of Squares, Cubes, Square and Cube Roots. 


Number. 

' 

Squares. 

I Cubes. 

Roots. 

Roots. 

Reciprocals. 

1353 

1830609 

2476813977 

36-7831483 

11-0602752 

•0007390983 

1354 

1833316 

2482309864 

36-7967390 

11 "0629994 

•0007385524 

1355 

1836025 

2487813875 

36-8103246 

11-0657222 

•0007380074 

1356 

1838736 

2493326016 

36-8239053 

11-06S4437 

-0007374631 

1357 

1841449 

2498846293 

36*8374809 

11-0711639 

-0007369197 

1358 

1S44164 

2504374712 

36-8510515 

11-0738828 

•0007363770 

1359 

1846881 

2509911279 

36-8646172 

11-0766003 

•0007358352 

1360 

1849600 

2515456000 

36-8781778 

11-0793165 

•0007352941 

1361 

1852321 

2521008881 

36-8917335 

11-0820314 

•0007347539 

1362 

1855044 

2526569928 

36-9052842 

11-0847449 

•0007342144 

1363 

1857769 

2532139147 

36-9188299 

11-0874571 

•0007336757 

1364 

1860496 

2537716544 

36-9323706 

11-0901679 

•0007331378 

1365 

1863225 

2543302125 

36-9459064 

11-0928775 

•0007326007 

1366 

1865956 

2548895896 

36-9594372 

11-0955857 

•0007320644 

1367 

1868689 

2554497863 

36-9729631 

11-0982926 

•0007315289 

1368 

1871424 

2560108032 

36-9864840 

11-1009982 

•0007309942 

1369 

1874161 

2565726409 

37-0000000 

11-1037025 

•0007304602 

1370 

1876900 

2571353000 

37-0135110 

11-1064054 

•0007299270 

1371 

1879641 

2576987811 

37-0270172 

11-1091070 

•0007293946 

1372 

1882384 

2582630848 

37-0405184 

11-1118073 

•0007288680 

1373 

1885129 

2588282117 

37-0540146 

11-1145064 

•0007283321 

1374 

1887876 

2593941624 

37-0675060 

11-1172041 

•0007278020 

1375 

1890625 

2599609375 

37-0899924 

11-1199004 

•0007272727 

1376 

1893376 

2605285376 

37-0944740 

11-1225955 

•0007267442 

1377 

1896129 

2610969633 

37-1079506 

11-1252893 

•0007262164 

1378 

1898884 

2616662152 

37-1214224 

11-1279817 

•0007256894 

1379 

1901641 

2622362939 

37-1348893 

11-1306729 

•0007251632 

1380 

1904400 

2628072000 

37-1483512 

11-1333628 

•0007246377 

1381 

1907161 

2633789341 

37-1618084 

11-1360514 

•0007241130 

1382 

1909924 

2639514968 

37-1752606 

11-1387386 

•0007235890 

1383 

1912689 

2645248887 

37-1887079 

11-1414246 

•0007230658 

13-84 

1915456 

2650991104 

37-2021505 

11-1441093 

•0007225434 

1385 

1918225 

2656741625 

37-2155881 

11-1467926 

•0007220217 

1386 

1920996 

2662500456 

37-2290209 

11-1494747 

•0007215007 

1387 

1923769 

2668267603 

37-2424489 

11-1521555 

•0007209805 

1388 

1926544 

2674043072 

37-2558720 

11-1548350 

•0007204611 

1389 

1929321 

2679826869 

37-2692903 

11-1575133 

•0007199424 

1390 

1932100 

2685619000 

37-2S27037 

11-1601903 

•0007194245 

1391 

1934881 

2691419471 

37-2961124 

11-1628659 

•0007189073 

1392 

1937664 

2697228288 

37-3095162 

11-1655403 

•0007183908 

1393 

1940449 

2703015457 

37-3229152 

11-1682134 

•0007178751 

1394 

1943236 

2708870984 

37-3363094 

11-1708852 

•0007173601 

1395 

1946025 

2714704875 

37-3496988 

11-1735558 

•0007168459 

1396 

1948816 

2720547136 

37-3630834 

11-1762250 

•0007163324 

1397 

1951609 

2726397773 

37-3764632 

11-1788930 

•0007158196 

1398 

1954404 

2732256792 

37-3898382 

11-1815598 

•0007153076 

1399 

1957201 

2738124199 

37-4032084 

11-1842252 

•0007147963 

1400 

1960000 

2744000000 

37-4165738 

11-1868894 

•0007142857 

1401 

1962801 

2749884201 

37-4299345 

11-1895523 

•0007137759 

1402 

1965604 

2755776808 

37-4432904 

11-1922139 

•0007132668 

1403 

1968409 

2761677827 

37-4566416 

11-1948743 

•0007127584 

1404 

1971216 

2767587264 

37-4699880 

11-1975334 

•0007122507 














50 Table of Squares, Cubes, Square and Cube Roots. 


Number. 

Squares. 

Cubes. 

y/ Roots. 

! * __ : r. 

V Roots. 

Reciprocals. 

1405 

1974025 

2773505123 

37*4833296 

11*2001913 

•000711743S' 

1406 

1976836 

2779431416 

37*4966665 

11*2028479 

•0007112376 

1407 

1979649 

2785366143 

37*5099987 

11*2055032 

•0007107321 

140S 

19S2464 

2791309312 

37*5283261 

1 1*2081573 

•0007102273 

1409 

1985281 

2797260929 

37*5366487 

11*2108101 

•0007097232 

1410 

1988100 

2803221000 

37*5499667 

11*2134617 

•0007092199 

1411 

1990921 

2809189531 

37*5632799 

11*2161120 

•0007087172 

1412 

1993744 

2815166528 

37*5765885 

11*2187611 

*0007082153 

1413 

1996569 

2821151997 

37*5898922 

11*2214089 

*0007077141 

1414 

1999396 

2827145944 

37*6031913 

11*2240054 

*0007072136 

1415 

2002225 

2833I48375 

37*6164857 

11*2267007 

*0007067138 

1416 

2005056 

2839159296 

37*6297754 

11*2293448 

*0007062147 

1417 

2007889 

2845178713 

37*6430004 

11*2319876 

*0007057163 

1418 

2010724 

2851206632 

37*6563407 

11*2346292 

*0007052186 

1419 

2013561 

2857243059 

37*6696164 

11*2372696 

*0007047216 

1420 

2016400 

2S63288000 

37*6828874 

11*2399087 

•0007042254 

1421 

2019241 

2869341461 

37*6961536 

11*2425465 

*0007037298 

1422 

2022084 

2875403448 

37*7094153 

11*2451831 

*0007032349 

1423 

2024929 

2881473967 

37*7226722 

11*2478185 

*0007027407 

1424 

2027776 

2887553024 

37*7359245 

11*2504527 

•0007022472 

1425 

2030625 

2893640625 

37*7491722 

11*2530856 

•0007017544 

1426 

2033476 

2899736776 

37*7624152 

11*2557173 

•0007012623 

1427 

2036329 

2905841483 

37*7756535 

11*2583478 

•0007007708 

1428 

2039184 

2911954752 

37*7888873 

11*2609770 

•0007002801 

1429 

2042041 

2918076589 

37*8021163 

11*2636050 

•0006997901 

1430 

2044900 

2924207000 

37*8153408 

11*2662318 

•0006993007 

1431 

2047761 

2930345991 

37*8285606 

11*2688573 

*0006988120 

1432 

2050624 

2936493568 

37*8417759 

11*2714816 

•0006983240 

1433 

2053489 

2942649737 

87*8549864 

11*2741047 

•0006978367 

1434 

2056356 

2948814504 

37*8681924 

11*2767266 

•0006973501 

1435 

2059225 

2954987875 

37*8813938 

11*2793472 

•0006968641 

1436 

2062096 

2961169856 

37*8945906 

11*2819666 

•000696378S 

1437 

2064969 

2967360453 

37*9077828 

11*2845849 

*0006958942 

1438 

2067844 

2973559672 

37*9209704 

11*2872019 

•0006954103 

1439 

2070721 

2979767519 

37*9341538 

11*2898177 

•0006949270 

1440 

2073600 

29859.84000 

37*9473319 

11*2924323 

*0006941444 

1441 

2076181 

2992209121 

37*9605058 

11*2950457 

•0006939625 

1142 

2079364 

3098442888 

37*9736751 

11*2976579 

*000(^934813 

1443 

20S2249 

3001685307 

37*9868398 

11*3002688 

•0006930007 

1444 

20S5130 

3010936384 

38-0000000 

11*3028786 

•0006925208 

1445 

2088025 

3017196125 

38*0131556 

11*3054871 

*0006920415 

1446 

2080916 

3023464536 

38*0263067 

11 *3 OS 0945 

•0006915629 

1447 

2093809 

3029741623 

38*0394532 

11*3107006 

•0006910850 

1448 

2096701 

3036027392 

38*0525952 

11*3133056 

•000690607S 

1449 

2099601 

3042321849 

38*0657326 

11*3159094 

*0006901312 

1450 

2102500 

3048625000 

38*0788655 

11*3185119 

•0006890552 

1451 

2105401 

3054936851 

3S-0919939 

11*3211132 

*0006891799 

1452 

2108304 

3061257408 

38*1051178 

11*3237134 

•0006887052 

1453 

2111209 

3067586777 

38*1182371 

11*3263124 

•0006882312 

1454 

2114116 

3073924664 

38*1313519 

11*3289102 

•0006877579 

1455 

2117025 

3080271375 

38*1444622 

11*3315067 

-0006872852 

1456 

...___ 

2119936 

3086626816 

38*1575681 

11*3341022 

*0006868132 

— 


















Taels of Squares, Cubes, Square and Cube Roots. 5] 



1 





Number. 

Squares. 

Cubes. 

V Roots. 

v/ Roots. 

Reciprocals. 

1457 

242 2 8 49 

3092990993 

38-1706693 

11-3366964 

•00068634:2 

1458 

2125764 

3099363912 

38-1837662 

11 -3392894 

•0006S58711 

1459 

2128681i3105745579 

38" 1968585 

11-341 SSI 3 

•0000854010 

1469 

2131600 

3112136000 

38-2099403 

11-3444719 

•0006849315 

1461 

2134521 

31 18535181 

38-2230297 

1 1 -3470614 

•0006844027 

1462 

2137444 

31249 43128 

38-2301035 

11-3496497 

•0006839945 

1463 

2140369 

3131359847 

38-2491829 

11-3522368 

*0006835270 

1461 

2143296 

3137785344 

38-2622529 

11-3548227 

*0006830601 

1465 

2146225 

3144219625 

38-2753184 

11-3574075 

*0006825939 

1466 

2149156 

3150662696 

38-2383794 

1 1-3599911 

•0006821282 

1467 

2152089 

3157114563 

38-3014360 

11-3025735 

•0006816633 

1468 

2155024 

3163575232 

38-3144881 

1 1-3651547 

•0006811989 

1469 

2157961 

3170044709 

38-3275358 

11 -3677347 

•0006807352 

1479 

2160900 

3176523000 

38-3405790 

11-3703136 

•0006802721 

147! 

2163841 

3183010111 

38-3536178 

1 1-3728914 

•0006798097 

1472 

2166784 

3189500048 

38-3606522 

1 1-3754679 

*0006793478 

1473 

2169729 

3196010817 

38-3796821 

1 1-3780433 

*0006788806 

1474 

2172676 

3202524424 

38-3927070 

11-3806175 

•0006784261 

1475 

2175625 

3209040875 

38 A 057287 

1 1-3831906 

•0006779661 

1476 

2178-576 

3215578176 

38-4187454 

1 1-3857625 

*0006775068 

1477 

2181529 

3222118333 

38-4317577 

11 -3883332 

•0006770481 

1478 

2184484 

3228607352 

38*4117050 

11-3909028 

*0006765900 

1479 

2187441 

3235225239 

38-4577091 

11-3934712 

•0006761325 

1489 

2190400 

3241792000 

38-4707081 

1 1-3960384 

‘0006756757 

1481 

2193361 

3248307641 

38-4837027 

1 1-3986045 

•0006752194 

1482 

2196324 

3254952108 

38-4967530 

1 1-4011695 

•0006747038 

1 483 

2199289 

3261545587 

38-5097390 

1 1-4037332 

*0006743088 

1484 

2202256 

3268147904 

3S-5227200 

1 1-4062959 

*0006738544 

14S5 

2205225 

3274759I25 : 

38-5350977 

1 1-4088574 

•0006734007 

1486 

2208196 

3281379256 

3S-5 480705 

11-4114177 

*0006729474 

1487 

2211169 

3288008303 

38-5010389 

1 1-4139769 

•0006724950 

1488 

2214144 

3294640272 

38-5740030 

11-4165349 

•0006720430 

1489 

2217121 

3301293109 

38-5S75627 

1 1 -4 1909 IS 

•0006715917 

1499 

2220100 

3307949000 

38-60051 SI 

11-4206476 

*0006711409 

1491 

2223081 

3314613771 

38-0134691 

11 -4242022 

•0006706908 

1492 

2226001 

3321287488 

38-6264158 

11-4267556 

•0006702413 

1493 

2229049 

3327970157 

38-6393582 

11-4293079 

•0000097924 

1494 

2232036 

3334661784 

38-6522962 

11 *4318591 

•0000698440 

1495 

2235025 

3341362375 

38-6052299 

11-4344092 

*0006088963 

1496 

2238016 

3348071936 

38*6781593 

11-436 9 5 SI 

*0000084492 

1 497 

2241009 

3354790473 

38-6910843 

11-4395059 

*0000680027 

1 198 

2244004 

3361517992 

38-7040050 

11-4420525 

*0000675667 

1499 

2247001 

3363254499 

38-7109214 

11-44459S0 

•0000671114 

1509 

2250000 

3375000000 

38-7298335 

11-4471424 

•0006666667 

1501 

2253001 

3381754501 

38-7427412 

11-4490857 

•0000662225 

1502 

22560041 

3388518008 

38-7550447 

11-4522278 

•0006657790 

1503 

2259009 

3395290527 

38-7685439 

11-45476SS 

*0006553360 

1504 

2262016 

3402072004 

38-7814389 

11-4573087 

•0000648936 

1505 

2265025 

3408862625 

38-7943294 

1 1-4598 476 

•0000644518 

1596 

2268036 

3415662216 

38-807215S 

11-4623850 

•0006640106 

1507 

2271049 

3422470843 

38-8200978 

11-4049215 

"0006635700 

1508 

2274064 

3429288512 1 

3S-8329757 

11-46745OS 

*0006631300 


L 

























62 Table of Squares, Cubes, Square and Cube Roots. 


Number. 

Squares. 

Cubes. 

vAKoots. 

•^Alloots. 

1 

Reciprocals. 

1509 

2277081 

3436115229 

38-8458491 

11-4699911 

•0006626905 

1510 

2280100 

3442951000 

3S-8587184 

11-4725242 

•0006622517 

1511 

2283121 

3449795831 

38*8715834 

11-4750562 

•0006618134 

1512 

2286144 

3456649728 

38-8844442 

11-4775871 

•0006613757 

1513 

2289169 

3463512697 

38-8973006 

11-4801169 

•00066093851 

1514 

2292196 

3470384744 

38-9101529 

11-4826455 

•0006605020 

1515 

2295225 

3477265875 

38-9230009 

11-4851731 

•0006600660 

1516 

2298256 

3484156096 

38-9358447 

11-4876995 

•0006596306 

1517 

2301289 

3491055413 

33-9486841 

11-4902249 

•0006591958 

1518 

2304324 

3597963832 

38-9615194 

11-4927491 

•0006587615 

1519 

2307361 

3504881359 

38-9743505 

11-4952722 

•0006583278 

1520 

2310400 

3511808000 

38-9871774 

11-4977942 

•0006578947 

1521 

2313441 

3518743761 

39-0000000 

1 i-5003151 

•0006574622 

1522 

2316484 

3525688648 

39-0128184 

11-5028348 

•0006570302 

1523 

2319529 

3532642667 

39-0256326 

11-5053535 

•0006565988 

1524 

2322576 

3539605824 

39-0384426 

11-5078711 

•0006561680 

1525 

2325625 

3546578125 

39-0512483 

11-5103876 

•0006557377 

1526 

2328676 

3553559576 

39-0640499 

11-5129030 

•0006553080 

1527 

2331729 

3567549552 

39-0768473 

11-5154173 

•0006548788 

1528 

2334784 

3560558183 

39-0896406 

11-5179305 

•0006544503 

1529 

2337841 

3574558889 

39-1024296 

11-5204425 

•0006540222 

1530 

2340900 

3581577000 

39-1152144 

11-5229535 

•0006535948 

1531 

2343961 

3588604291 

39-1279951 

11-5254634 

•0006531679 

1532 

2347024 

3595640768 

39-1407716 

11-5279722 

•0006527415 

1533 

2350089 

3602686437 

39-1535439 

11-5304799 

•0006523157 

1534 

2353156 

3609741304 

39-1663120 

11-5329865 

•0006518905 

1535 

2356225 

3616805375 

39-1790760 

11-5354920 

•0006514658 

1536 

2359256 

3623878656 

39-1918359 

11-5379965 

•0006510417 

1537 

2362369 

3630961153 

39-2045915 

11-5404998 

•0006506181 

1538 

2365444 

3638052872 

39-2173431 

11-5430021 

•0006501951 

1539 

2368521 

3645153S19 

39-2300905 

1 1-5455033 

•0006497726 

1540 

2371600 

3652264000 

39-2428337 

11-5480034 

•0006493506 

1541 

2374681 

3657983421 

39-2555728 

11-5505025 

•0006489293 

1542 

2377764 

3666512088 

39-2683078 

11-5530004 

•0006485084 

1543 

2380849 

3673650007 

39-2810387 

11-5554972 

•0006480881 

1544 

2383936 

3680797184 

39-2937654 

11-5579931 

•0006476684 

1545 

2387025 

3687953625 

39-3064880 

11-5604878 

•0006472492 

1546 

2390116 

3695119336 

39-3192065 

11-5629815 

•0006468305 

1547 

2393209 

370229^323 

39-3319208 

11-5654740 

•0006464124 

1548 

2396304 

3709478592 

39-3446311 

11-5679655 

•0006459948 

1549 

2399401 

3716672149 

39-3573373 

11-5704559 

•0006455778 

1550 

2402500 

3723875000 

39-3700394 

11-5729453 

•0006451613 

1551 

2405601 

3731087151 

39-3827373 

11-5754336 

•0006447453 

1552 

2408704 

373830S608 

39-3954312 

11-5779208 

•0006443299 

1553 

2411809 

3745539377 

39-4081210 

11-5804069 

•0006439150 

1554 

2414916 

3752779464 

39-4208067 

11-5828919 

•0006435006 

1559 

2418025 

3760028875 

39-4334883 

11-5853759 

•0006430868 

] 556 

2421136 

3767287616 

39-4461658 

11-5878588 

•0006426735 

1557 

2424249 

3774555693 

39-4588393 

11-5903407 

•0006422608 

1558 

2427364 

3781833112 

39-4715087 

11-5923215 

•0006418485 

1559 

2430481 

3789119879 

39-4841740 

11-5953013 

•0006414368 

1560 

2433600 

37964160001 

39-4968353 

11-5977799 

•0006410256 


















Table of Squares, Cubes, Square and Cube Roots. 53 




j 

- 1 



Number 

Squares. 

Cubes. 

vRoots. 

Roots. 

Reciprocals. 

1561 

2436721 

3803721481 

39-5094925 

11 6002576 

•0006406150 

1562 

2439844 

3811036328 

39-5221457 

11-6027342 

•0006402049 

1563 

2442969 

3818360547 

39-5347948 

11-6052097 

•0006397953 

1504 

2446096 

3825641444 

39-5474399 

11-6976841 

•0006393862 

1565 

2449225 

3833037125 

39-5600809 

H'6101575 

•0006389776 

1566 

2452356 

3840389496 

39-5727179 

11-6126299 

•0006385696 

1567 

24554S9 

3847751263 

39-5853508 

11-6151012 

•0006381621 

1568 

2458624 

3855123432 

39-5979797 

1 1-6175715 

•0006377551 

1569 

24C1761 

3862503009 

39-6106046 

11-6200407 

•0006373486 

1570 

2464900 

3869883000 

39-6232255 

11-6225088 

•0006369427 

1571 

2468041 

3877292411 

39-6358424 

11-6249759 

•0006365372 

1572 

2471184 

3884701248 

39-6484552 

11-6274420 

•0006361323 

1573 

2474329 

3892119157 

39-6610640 

11-6299070 

•0006357279 

1574 

2477476 

3899547224 

39-6736688 

11 -6323710 

•0006353240 

1575 

2480625 

3906984375 

39-6862096 

11-6348339 

•0006349206 

1576 

2483776 

3914430976 

39-6988665 

11-6372957 

•0006345178 

1577 

2486929 

3921887033 

39-7114593 

11-6397566 

•0006341154 

1578 

2490084 

3929352552 

39-7240481 

11-6422164 

•0006337136 

1579 

2493241 

3936827539 

39-7306329 

11-6446751 

•0006333122 

1580 

2496400 

3944312000 

39-7492138 

11-6471329 

•0006329114 

1581 

2499561 

3951S05941 

39-7617907 

11-6495895 

•0006325111 

1582 

2502724 

3959309368 

39-7743636 

11-6520452 

•0006321113 

1583 

2505889 

3966822287 

39-7869325 

11-6544998 

•0006317119 

15S4 

2509056 

3974344704 

39-7994976 

11-6569534 

•0006313131 

1585 

2512225 

3981876625 

39-8120585 

11-6594059 

•0006309148 

1586 

2515396 

3989418056 

39-8246155 

11-6618574 

•0006305170 

1587 

2518569 

3996969003 

39-8371686 

11-6643079 

•0006301197 

1588 

2521744 

4004529472 

39-8497177 

11-6667574 

•0006297229 

15S9 

2524921 

4012099469 

39-8622628 

11-6692058 

•0006293266 

1590 

2528100 

4014679000 

39-8748040 

11-6716532 

•00062S9308 

1591 

25312S1 

4027-268071 

39-8873413 

11-6740996 

•0006285355 

1592 

2534464 

4034866688 

39-8998747 

11-6765449 

•0006281407 

1593 

2537649 

4042474857 

39-9124041 

11-6789892 

•0006277464 

1594 

2540S36 

4050092584 

39-9249295 

11-6814325 

•0006273526 

1595 

2544025 

4057719S75 

39-9374511 

11-683S748 

•0006269592 

1596 

2547216 

4065356736 

39-9499687 

11-6863161 

•0006265664 

1597 

2550409 

4073003173 

39-9624824 

11-6S87563 

•0006261741 

1598 

2553604 

4080659192 

39-9749922 

11-6911955 

•0006257822 

1599 

2556S01 

40S8324799 

39-9874980 

11-6936337 

•0006253909 

1600 

2560000 

4096000000 

40-0000000 

11-6960709 

•0006250000 


To find tht Square Root of Numbers exceeding 1C00. 

Example 4. Require the Square Root of 34C9S. In the column of Squares you 
will find, 

+34969 «= 1ST 2 , -f 34969 = 1ST 2 , 

—3469S = 186*+... —34596 = ISC*. 

~~m divided by ‘ 373"= 000.727 . 

^34698 = 186*727 nearly. 





















54 


Evolution. 


When the number contains Integer and Decimals. 

Example 5. Required the Square Root of 7845*45? In the column of Squares 


you will find, 


+7849*96 = 88-62, 
—7845-45 == 88-52-, 

451 divided by 


4-7849-96 = 88-62, 

—7832*25 = 88*52, 

"1771 = 00-0256. 
y/7845*45 = 88-5256 nearly. 


4®=*When the number of ciphers in the integer is even, the number of 
ciphers taken in the Square column must also be even ; but when the number 
of ciphers in the integer is odd, the number taken in the Square column must 
also be odd. 

To find the Cube Root of Numbers exceeding 1600. 

Example 6. Required the Cube Root of 5694958 ? In the Cube column you will 
find, 

4-5735339 = 1793 4-5735339 = 1793. 

—5694958 = 1783- — 5639752 = 1783. 

95587 = 000-4225, 

^5694958 = 178-4225 nearly. 


40381 divided by 


When the number contains Integer and Decimals. 

Example 7. Required the Cube Root of 4186-586? In the column of Cubes you 
will find, 

4-4251-528 = 16-23 4251-528 = 18*2* 

—4186-585 = 16-13- 4173-281 = 16-13 


64942 


78247 = 00-083 


4186-586 = 16-183 nearly. 


jgS^The following notice must be particularly attended to, when extracting 
Cube Root of numbers with decimals. 

2 ciphers in the integer must be 5, 8, or 11 ciphers in the Cube column. 


3, 6, or 9 

4, or 7 

5, or 8 

6, or 9 

7, or 10 


Example 8. Required the Cube Root of 61358-75? 
ciphers you will find, 


In the Cube column and 8 


4-61629-875 = 3953 
— 61358-750 = 3943- 

271-125 divided by 


4-61629875 = 39-53 
—61162984 = 39-43 
466891 = 00-05807 
61358-75 = 39-45807. 


To find the Fourth Root. 

Ride. Extract the Square Root of the number as before described, and of 
that root extract the Square Root again, then the last is the Fourth root of 
the number. 

Example 9. Required the fourth root of 2469781 ? 


2469781 = V )/2469781 = ]/1571-4463 = 39 - 6467, the answer. 

To find the Sixth Root. 

Rule. Find the Cube Root of the number as before described, and of that 
root extract the Cube Root again, and then the last is the Sixth root of the 
number. 


.1 





















Nystrom’s Calculator. 


55 



NYSTROM’S CALCULATOR. 



All calculations in this Pocket Book have been computed by this Instrument. 
It consists of a silvered brass plate on which are fixed two moveable arms, ex¬ 
tending from the centre to the periphery. On the plate are engraved a number 
of curved lines in such form and divisions, that by their intersection with the 
arms, numbers are read and problems solved. 

The arrangement for trigonometrical calculations is such that it is not neces¬ 
sary to notice sine, cosine, tan, &c., &c., operating only by the angles them¬ 
selves expressed in degrees and minutes. This makes trigonometrical solutions 
so easy, that any one who understands Simple Arithmetic, will be able to solve 
trigonometrical questions. 

Calculations are performed by it almost instantly, no matter how complicated 
they may be, while there is nothing intricate or difficult in its use. The author 
of this book, who is the inventor, has in the last five years thoroughly tested its 
practical utility. 

ADVERTISEMENT. 

The attention of Engineers, Ship builders, and all whose business requires 
frequent and extensive calculations, is called to Nystrom’s Calculator. Price $15. 
To be obtained with description by applying to John W. Nystrom, Philadelphia. 

Communications will be promptly attended to. 

This Calculator received the First Premium at the Franklin Institute 
Exhibition. 

GEORGE THORESTED, Manufacturer, 

23 Hammersly Street, 
New York. 































Lojaeithms. 


to 


L 0 G A R I T II M S 


A Logaritlun is an exponent of a power to which 10 must he raised to give 
a certain number, which will be understood by this 


c 

3 

o' 


los. 100 


o 

05 

p 


tx 

S 




log. 10000 = 4 
log. 5012 


. x 

~ O 
0 

<T> 

0 


because 10 2 = 100. 

“ 10 4 = 10000 


oU 


103-7 = 5012. 


The unit of the logarithm is called the characteristic or index , and the decimal 
part is called the mantissa, the sum of the characteristic and mantissa is the Loga¬ 
rithm. The invariable number 10 is the base for the system of Loga¬ 
rithms. 

It is not necessary that the base should be 10, it can be any number, but all 
the tables of logarithms now in common use, are calculated with 10 to the 
base. 

The nature of logarithms in connection with their numbers are such, that the 
index of the logarithm is always one less than the number of figures in the 
number, (when the base of the logarithm is 10,) as, 

index 5012 = 3 
mantissa 5012 = 0'7 

logarithm 5012 = 3 - 7. 

Let 10 be raised to any power x, and 

the power of 10* = a or log. a = x, 
the power of 10® = b or log. b — z. 

Let the product of ab = c and the quotient-^ = c. 


10*X10 S = 10** z = ab = c 

10* a 

-— =10*—* = — =d 
10* b 


or log. c — x+z. 
or log. d — x — z. 


a = m z or log. m = -zXlog. a. 

a = n or log. n = log. a : 3. 

Any number represented by the letters a, b, c, or d, can be a power of 10, which 
exponent is the logarithm for the number. Logarithms are calculated for every 
number with three figures in the accompanying Table, by which any operation in 
Multiplication. Division, Involution and Evolution can be performed by simple 
Addition or Subtraction of Logarithms. Tables of Logarithms are commonly more 
extensive, and calculated tor a.ny number of four or five figures, which would 
occupy too much room in this book; but by the proportional parts, the logarithm 
can be found by this Table, to four or five figures. The index of the logarithms 
do not appear in the Table, only the mantissa. It is easily remembered that the 
index is one less, than the number of figures in the number; then when the num¬ 
ber is only one figure, the index is 0 ; and when the number is a fraction, the 
index is negative. 

When the logarithm is to be found for a fraction, we commonly have the 
fraction expressed in a decimal; and then the negative index is equal to the 
number of ciphers before the first figure, and commonly marked after the man¬ 
tissa; thus explained in whole numbers and fractions: 

log. 365 = 2-56229.. log. 0-365 = '56229—1. 
log. 46-7 = 1-66931 log. 0-0467 = -66931—2. 
log. 7-59 = 0-88024 log. 0-00759 = -88024—3. 

In the accompanying Table of Logarithms, for the trigonometrical lines the 
negative index is marked thus, 

log. sin. 35° 40' = log. 0-58306 = 1:76572. 















Logarithms. 


57 


To find the Logarithm of Numbers. 

Example 1. Find the logarithm of 45 ? 

To 45 in the first column of the Table, answers 65321 in the next column, 
which is the mantissa; index = 1 because 45 is two figures. 

Then, log. 45 == T65321, the answer. 

Example 2. Find the logarithm of 768 ? 

Opposite 76 in the first column, answers 88536 in the column marked 8 on the 
top or bottom. Index = 2 because 768 is three figures. 

Then, log. 768 =*= 2-88536. 

Example 3. Find the Logarithm of 6846 ? 

log. 6840 = 3-83505 
Proportional part, 64X0-6 = 384 

log. 6846 = 3'835434 the answer. 

To find the number for a given Logarithm. 

Example 1. What number answers to the logarithm 3-87157 ? 

In the Table you will find in the column of logarithms, that 

log. 7440 = 3-87157. 

Example 2. What number answers to the logarithm 3-801884? 

Given logarithm 3'801884, 

Subt. nearest table log. 3-801400 = log. 6330, 

Divided by proportional part, 69 |4S4| - - - - - - 7, 

6337 the req. numb. 

Multiplication by Logarithms. 

Rule. Add together the logarithms of the factors, and the sum is the loga¬ 
rithm of the product. 

Example 1. Multiply 425 by 48. 

To log. 425 = 2-62839, 

Add log. 48 = 1-68124, 

The product, log. 20400 = 4-30963. 

Example 2. Multiply 79600 by 0-435. 

To log. 79600 = 4-90091, 

Add, log. 0-435 = -63848—1, 

The product log. 34090 = 4-53939. 

Division by Logarithms. 

Rule. From the logarithm of the dividend subtract the logarithm of the di¬ 
visor, and the difference is the logarithm of the quotient. 

Example 1. Divide 43800 by 368. 

From log. 43800 = 4-64147, 

Subtract log. 368 = 2-56584, 

The quotient log, 119 = 2-07563. 

Example 2. Divide 36 by 0.625, 

From log. 36 = 1-55636, 

Subtract, log. 0-625 = ‘79588-1. 

The quotient, log. 57-6 = 1-76048. 

A negative index follows an opposite operation of its mantissa, as if the man¬ 
tissa is subtracted, add the negative index, and vice versa. 

Envolution by Logarithms. 

Rule. Multiply the logarithm of the number by its exponent, and the pro¬ 
duct is the logarithm of the power of the number. 

Involution by Logarithms. 

Rule. Divide the logarithm of the number by the index of the root, and the 
quotient is the logarithm of the root of the number. 









58 


Logarithm of Numbers. 


LOGARITHM OF NUMBERS, FROM I 


O to 1000, 


No. 

0 

1 

2 

o 

O 

4 

5 

6 

7 

8 

9 

Prop. 

0 

0 

00000 

30103 

47712 

60206 

69897 

77815 

84510 

90309 

95424 


10 

00000 

00432 

00S60 

01233 

01703 

02118 

02530 

0293S 

03342 

03742 

115 

11 

04139 

04532 

04921 

05307 

05090 

06069 

06445 

06818 

07188 

07554 

379 

12 

07918 

08278 

08636 

08990 

09342 

09691 

10037 

10380 

10721 

11059 

349 

13 

11394 

11727 

12057 

12385 

12710 

13033 

13353 

13672 

13987 

14301 

323 

14 

14612 

14921 

15228 

15533 

15836 

16136 

16435 

16731 

17026 

17318 

300 

15 

17609 

17897 

18184 

18469 

18752 

19033 

19312 

19590 

19865 

20139 

2S1 

16 

20412 

20682 

20951 

21218 

21484 

21748 

2201 ( 

22271 

22530 

22788 

204 

17 

23044 

23299 

23552 

23S04 

24054 

24303 

24551 

24797 

25042 

25285 

249 

18 

25527 

25767 

26007 

26245 

26481 

26717 

26951 

27184 

27415 

27646 

236 

19 

27875 

28103 

28330 

28555 

28780 

29003 

29225 

29446 

29660 

29885 

223 

20 

30103 

30319 

30535 

30749 

30963 

31175 

31386 

31597 

31806 

32014 

212 

21 

32221 

32428 

32633 

32838 

33041 

33243 

33445 

33646 

33845 

34044 

202 

22 

34242 

34439 

34035 

34830 

35024 

35218 

35410 

35602 

35793 

35983 

194 

23 

36172 

36361 

36548 

36735 

36921 

37106 

3 72 91137474 

37657 

37839 

185 

24 

38021 

38201 

38381 

38560 

38739 

38916 

39093:39269 

40824140993 

39449 

39619 

177 

25 

39794 

39967 

10140 

40312 

40483 

40654 

11162 

41330 

171 

26 

41497 

41664 

41830 

41995 

42160 

42324 

12488 

42651 

12813 

42975 

164 

27 

13136 

43296 

43456 

43616 

43775 

43933 

44090 

44248 

14404 

44560 

158 

28 

44715 

44870 

15024 

45178 

45331 

15484 

15630 

45788 

15939 

46089 

153 

29 

46239 

46389 

46538 

16686 

16834 

46981 

17129 

47275 

17421 

17567 

148 

30 

47712 

17856 

18000 

48144 

48287 

48430 

18572 

48713 

18855 

18995 

143 

31 

49136 

19276 

49415 

19554 

49693 

19831 

49968 

50105 

50242 

50379 

138 

32 

50515 

50650 

50785 

50920 

51054 

51181 

51321 

51454 

51587 

51719 

134 

33 

51851 

51982 

52113 

52244 

52374 

52504 

52633 

52763 

52891 

53020 

130 

34 

53147 

53275 

53402 

53521 

53655 

53781 

53907 

54033 

54157 

54282 

126 

35 

54406 

54530 

54654 

54777 

54900 

55022 

55145 

55266 

,35388 

55509 

122 

36 

55630 

55750 

55870 

55990 

56110 

56229 

56348 

56466 

56584 

56702 

119 

37 

56820 

56937 

57054 

57170 

57287 

57403 

57518 

57634 

57749 

57863 

116 

38 

57978 

58092 

58206 

58319 

58433 

58546 

58658 

58771 

58883 

58995 

113 

39 

59106 

59217 

59328 

59439 

59549 

59659 

59769 

59879 

59988 

60097 

110 

40 

60206 

80314 

60422 

60530 

80638 

60745 

60852 

60959 

61066 

61172 

107 

41 

61278 

61384 

61489 

61595 

61700 

61804 

61909162013 

62117 

62221 

104 

42 

32324 

62428 

62531 

62634 

62736 

62838 

62941 

63042 

63144 

63245 

102 

43 

33346 

53447 

63548 

63648 

63749 

63848 

63948i6104S 

64147 

64246 

99 

44 

54345 

64443 

61542 

64640 

84738 

64836 

64933 65030 
65890165991 

65127 

65224 

98 

45 

55321 

65417 

65513 

65609 

65705 

65801 

66086 

66181 

96 

46 

36275 

66370 

66464 

66558 

66651 

66745 

66838 

66931 

67024 

67117 

94 

47 

57209 

67302 

67394 

67486 

67577 

67669 

67760 

67851 

67942 

68033 

92 

48 

38124 

68214 

68304 

68394 

68484 

68574 

68665 

6S752 

68842 

68930 

90 

49 

59019 

69108 

69196 

69284 

69372 

69460 

69548 

69635 

69722 

69810 

88 

50 

69897 

69983 

70070 

70156 

70243 

70329 

70415 

70500 

70586 

70671 

86 

51 

70757 

70842 

70927 

71011 

71096 

71180 

71265 

71349 

714.33 

71516 

84 

52 

71600 

71683 

71767 

71850 

71933 

72015 

7209 S 

72181 

72263 

71315 

82 

53 

72427 

72509 

72591 

72672 

72754 

72835 

72916 

72997 

73078 

73158 

81 

54 

73239 

73319 

73399 

73480 

73559 

73639 

73719 

73798 

73878 

73957 

80 

55 

No. 

74036 

0 

74115 

1 

74193 

2 

74272 

. 3 

74351 

74429 

5 

74507 

6 

74585 

74663 

L«j 

74771 

9 

78 

Prop 

L 4 

7 




























Logarithms of Numbers. 


59 


No. 

u 1 

1 

2 

3 

4 

5 

6 

7 

-- 

8 

9 

Prop. 

56 

748 IS 

74896 

74973 

75050 

75127 

75204 

75281 

75358 

75434 75511 

77 

57 

75587 

75663 

7 5739 

75S15 

75891 

75966 

76042 

76117 

76192 

76267 

75 

5 b 

1 6342 

76417 

76492 

7 6566 

76641 

76713 

76789 

76863 

76937 

77011 

74 

59 

7 70 8 i 

7715b 

77232 

77305 

77378 

77451 

77524 

77597 

77670 

77742 

73 

66 

77815 

77887 

77959 

78031 

78103 

7 817 5 

78247 

78318 

78390 

78461 

72 

61 

78533 

78604 

78675 

78746 

78816 

78887 

7S95S 

79028 

79098 

79165 

71 

62 

79239 

79309 

7 9379 

79448 

79518 

7958b 

79657 

79726 

79790 

79865 

70 

63 

79934 

80002 

80071 

80140 

80208 

80277 

80345 

80413 

50482 

80550 

69 

64 

1061c 

80685 

80753 

80821 

80888 

80956 

81023 

81090 

51157 

81224 

6S 

65 

11291 

81358 

81424 

81491 

31557 

31624 

SI 690 

81756 

81822 

8188b 

67 

66 

31954 

82020 

32085 

82151 

82216 

82282 

82347 

52412 

82477 

82542 

66 

67 

82607 

82672 

82736 

32801 

82866 

82936 

82994 

83058 

83123 

83187 

65 

68 

83250 

83314 

83378 

33442 

83505 

83569 

83632 

83695 

8375b 

83821 

64 

69 

83884 

S3947 

84010 

84073 

84136 

8419b 

84260 

84323 

84385 

54447 

63 

70 84509 

84571 

84633 

S 1695 

84751 

3481b 

84880 

84941 

85003 

55064 

62 

71 

3512c 

S51S7 

85248 

35309 

85369 

35436 

S5491 

85551 

85612 

85672 

61 

72 

85733 

85793 

85853 

35913 

35973 

86033 

86093 

86153 

86213 

86272 

60 

7 o 

86332 

83991 

36451 

86510 

36569 

86628 

86687 

86746 

86805 

86864 

59 

74 

38923 

86981 

37040 

87098 

37157 

37215 

87273 

87332 

37390 

87448 

58 

75 

87500 

87564 

87621 

37679 

87737 

37794 

87852 

37906 

87960 

88024 

57 

70 

88081 

8813b 

88195 

88252 

88309 

88366 

88422 

58479 

88530 

88592 

56 

77 

88619 

bb7 Oil 

88761 

88818 

38874 

88936 

88986 

89042 

8909 b 

39158 

56 

7 b 

19209 

89265 

89320 

89376 

89431 

89487 

89542 

89597 

59652 

39707 

55 

79 

89762 

89817 

89872 

39927 

89982 

90036 

90091 

40145 

90200 

40254 

54 

80 

90309 

90363 

90417 

90471 

90525 

90579 

90633 

90687 

40741 

40794 

54 

81 

9084c 

90902 

90955 

91009 

91062 

91115 

91169 

91222 

91275 

9132b 

53 

82 

91381 

91434 

91487 

91540 

91592 

91645 

91698 

91750 

91805 

91855 

53 

83 

91907 

91960 

92012 

92064 

92116 

9216b 

92220 

92272 

42324 

92370 

52 

84 

92427 

92479 

92531 

92582 

92634 

92685 

92737 

92788 

92839 

92890 

51 

85 

929 41 

92993 

93044 

93095 

9317 6 

93196 

93247 

93298 

9337b 

93399 

51 

80 

93449 

93500 

93550 

93601 

93651 

93701 

93751 

43802 

93852 

93902 

50 

87 

93951 

94001 

94051 

94101 

94151 

94206 

94250 

94300 

44349 

9439b 

49 

Sb 

9444b 

94497 

94546 

94596 

94645 

94694 

94743 

94792 

44841 

44890 

49 

89 

94939 

94987 

95036 

95085 

95133 

95182 

95230 

95279 

95327 

95376 

48 

90 

95424 

95472 

95520 

95568 

95616 

95664 

95712 

95760 

9580b 

45856 

48 

91 

95984 

95951 

95999 

56047 

96094 

96142 

96189 

96236 

96284 

46331 

48 

92 

96378 

96426 

96473 

98520 

96567 

96614 

96661 

46708 

96754 

96801 

47 

93 

96S4b 

96895 

96941 

96988 

97034 

97081 

97127 

97174 

47220 

47266 

47 

9-i 

97312 

97359 

97405 

97451 

97497 

97543 

97589 

97635 

97680 

47726 

56 

9u 

97772 

9781b 

97863 

97909 

97954 

93000 

98045 

98091 

48130 

48181 

56 

90 

98227 

98272 

98317 

98362 

98407 

98452 

98497 

98542 

48587 

9S632 

55 

97 

98677 

98721 

98766 

98811 

98855 

98901 

9S945 

48989 

99033 

9907b 

55 

9b 

99121 

99166 

99211 

99255 

99299 

99343 

99387 

99431 

99475 

99519 

54 

| 9'j!99560 

99607 

99651 

99694 

99738 

99782 

99825 

99869 

99913 

99956 

54 

1 Y., 

1 

1 0 

* 1 

Q 

3 

4 

5 

6 

1 

, 8 

9 

'^rop. 


Examples. Find the Logarithm of 

Log; 3 ? - 

Log. 54? - 

Log.867 ? 


G'47712, the answer. 

1- 73239, “ 

2- 93802, “ 


f 






























60 


Logarithms Sine. 


- t 

Deg, 

0' 

10' 

20' 

30' 

40' 

50' 

60' 

rr——“ 

0 

0:00000 

3:46372 

3:26475 

3:94084 

2:06577 

2:16268 

2:24185 

89 

1 

2:24185 

2:30879 

2:36677 

2:41791 

2:46366 

2:50504 

2:54281 

88 

2 

2:54281 

2:57756 

2:60973 

2:63968 

2:66768 

2:69399 

2:71880 

87 

3 

2:71880 

2:74225 

2:76451 

2:78567 

2:80585 

2:82513 

2:84358 

86 

4 

2:84358 

2:86128 

2:87828 

2:89461 

2:91040 

2:92560 

2:94029 

85 

5 

2:94029 

2:95449 

2:96824 

2:98157 

2:99449 

1:00704 

1:01923 

84 

6 

1:01923 

1:03108 

1:04262 

1:05385 

1:06480 

1:07548 

1:08589 

83 

t 

1:08589 

1:09606 

1:10599 

1:11569 

1:12518 

1:13447 

1:14355 

82 

8 

1:14355 

1:15249 

1:16116 

1:16970 

1:17807 

1:18628 

1:19433 

81 

9 

1:19433 

1:20223 

1:20999 

1:21760 

1:22509 

1:23244 

1:23967 

80 

10 

1:23967 

1:24677 

1:25376 

1:26063 

1:26739 

1:27404 

1:28059 

79 

11 

1:28059 

1:28704 

1:29339 

1:29965 

1:30581 

1:31189 

1:31787 

78 

12 

1:31787 

1:32378 

1:32959 

1:33533 

1:34099 

1:34657 

1:35208 

77 

13 

1:35208 

1:35752 

1:36288 

1:36818 

1:37341 

1:37857 

1:38367 

76 

14 

1:38367 

1:38871 

1:39368 

1:39860 

1:40345 

1:40825 

1:41299 

75 

15 

1:41299 

1:41768 

1:42231 

1:42689 

1:43142 

1:43590 

1:44033 

74 

16 

1:44033 

1:44472 

1:44905 

1:45334 

1:45758 

1:46178 

1:46593 

73 

17 

1:46593 

1:47004 

1:47411 

1:47814 

1:48212 

1:48607 

1:48998 

72 

18 

1:48998 

1:49385 

1:49768 

1:50147 

1:50523 

1:50895 

1:51264 

71 

19 

1:51264 

1:51629 

1:51991 

1:52349 

1:52704 

1:53056 

1:53405 

70 

20 

1:53405 

1:53750 

1:54093 

1:54432 

1:54768 

1:55102 

1:55432 

69 

21 

1:55432 

1:55760 

1:56085 

1:56407 

1:56726 

1:57043 

1:57357 

68 

22 

1:57357 

1:57668 

1:57977 

1:58284 

1:58587 

1:58889 

1:59187 

67 

23 

1:59187 

1:59484 

1:59778 

1:60070 

1:60359 

1:60646 

1:60931 

66 

24 

1:60931 

1:61214 

1:61494 

1:61772 

1:62048 

1:62322 

1:62594 

65 

25 

1:62594 

1:62864 

1:63132 

1:63398 

1:63662 

1:63924 

1:64184 

64 

26 

1:64184 

1:64442 

1:64698 

1:64952 

1:65205 

1:65455 

1:65704 

63 

27 

1:65704 

1:65951 

1:66197 

1:66440 

1:66682 

1:66922 

1:67160 

62 

28 

1:67160 

1:67397 

1:67632 

1:67866 

1:68098 

1:68328 

1:68557 

61 

29 

1:68557 

1:68784 

1:69009 

1:69233 

1:69456 

1:69677 

1:69897 

60 

30 

1:69897 

1:70115 

1:70331 

1:70546 

1:70760 

1:70773 

1:71183 

59 

31 

1:71183 

1:71393 

1:71601 

1:71808 

1:72014 

1:72218' 

.1:72421 

58 

32 

1:72421 

1:72622 

1:72822 

1:73021 

1:73219 

1:73415 

1:73610 

t 57 

33 

1:73610 

1:73804 

1:73997 

1:74188 

1:74379 

1:74568 

1:74756 

56 

34 

1:74756 

1:74942 

1:75128 

1:75312 

1:75496 

1:75678 

1:75859 

55 

35 

1:75859 

1:76039 

1:76217 

1:76395 

1:76572 

1:76747 

1:76921 

54 

36 

1:76921 

1:77095 

1:77267 

1-77438 

1:77609 

1:77778 

1:77946 

53 

37 

1:77946 

1:78113 

1:78279 

1:78444 

1:78608 

1:78772 

1:78934 

52 

38 

1:78934 

1:79095 

1:79255 

1:79415 

1:79573 

1:79730 

1:79887 

f 51 

39 

1:79887 

1:80042 

1:80197 

1:80351 

1:80503 

1:80655 

1:80806 

50 

40 

1:80808 

1:80956 

1:81106 

1:81254 

1:81401 

1:81548 

1:S1694 

,49 

41 

1:81694 

1:81839 

1:81983 

1:82126 

1:82268 

1:82410 

1:82551 

1 48 

42 

1:82551 

1:82691 

1:82830 

1:82968 

1:83105 

1:83242 

1:83378 

47 

43 

1:83378 

1:83513 

1:83647 

1:83781 

1:83914 

1:84045 

1:84177 

| 46 

44 

1 :S4177 

1:84307 

1:84437 

1:84566 

1:84694 

1:84821 

1:84948 

45 

45 

1:84948 

1:85074 

1:85199 

1:85324 

1:8544S 

1:85571 

1:85693 

44 


60' 

50' 

40' 

30' 

1 20' 

10' 

L 0' 



Logarithm Cosine. 


The negative index is noted by two points, and mast always follow an oppo 
site operation to that of the mantissa. If the mantissa is added, subtract 
i the index, and vice versa. 

































Logarithms Sine. G1 


Deg. 

O' 

10 ' 

20 ' 

30' 

40' 

50' 

60' 


46 

1:85693 

1:S5S15 

1:85936 

1:86056 

1:86175 

1:86294 

1:86412 

43 

47 

1:86412 

1:86530 

1:86647 

1:86763 

1:86878 

1:86993 

1:87107 

42 

43 

1:87107 

1:37220 

1:87333 

1:87445 

1:87557 

1:87667 

1:87778 

41 

49 

1:87778 

1:87837 

1:87996 

1:88104 

1:88212 

1:88319 

1:88425 

40 

50 

1:88425 

1:88531 

1:S8636 

1:88740 

1:88844 

1:88947 

1:89050 

39 

51 

1:89050 

1:89152 

1:89253 

1:89354 

1:89454 

1:89554 

1:89653 

38 

52 

1:S9653 

1:89751 

1:89849 

1:89946 

1:90043 

1:90139 

1:90234 

37 

53 

1:90234 

1:90329 

1:90424 

1:90517 

1:90611 

1:90703 

1:90795 

36 

54 

1:90795 

1:90837 

1:90978 

1:91068 

1:91158 

1:91247 

1:91336 

35 

55 

1:91336 

1:91424 

1:91512 

1:94599 

1:91685 

1:91771 

1:91857 

34 

56 

1:91857 

1:91942 

1:92026 

1:92110 

1:92194 

1:92276 

1:92359 

33 

57 

1:92359 

1:92440 

1:92522 

1:92602 

1:92683 

1:92762 

1:92842 

32 

58 

1:92842 

1:92920 

1:92998 

1:93076 

1:93153 

1:93230 

1:93306 

31 

59 

4:95306 

1:93382 

1:93457 

1:93532 

1:93606 

1:93679 

1:93753 

30 

60 

1:93753 

1:93825 

1:93898 

1:93969 

1:94040 

1:9 i111 

1:94181 

29 

61 

1:94181 

1:94251 

1:94321 

1:94389 

1:94458 

1:94526 

1:94593 

28 

62 

1:94593 

1:94660 

1:94726 

1:94792 

1:9485S 

1:94923 

1:94988 

27 

63 

1:94988 

1:95052 

1:95115 

1:95179 

1:95241 

1:95304 

1:95366 

26 

64 

1:95366 

1:95427 

1:95488 

1:95548 

1:95608 

1:95668 

1:95727 

25 

65 

1:95727 

1:95786 

1:95844 

1:95902 

1:95959 

1:96016 

1:96073 

24 

66 

1:96073 

1:96129 

1:961 S4 

1:96239 

1:96294 

1:96348 

1:96402 

23 

67 

1:96402 

1:96456 

1:96509 

1:96561 

1:96613 

1:96665 

1:96716 

22 

68 

1:96716 

1:96767 

1:96817 

1:96867 

1:96917 

1:96966 

1:97015 

21 

69 

1:97015 

1:97063 

1:97111 

1:97158 

1:97205 

1:97252 

1:97298 

20 

70 

1:97298 

1:97344 

1:97389 

1:97434 

1:97479 

1:97523 

1:97567 

19 

71 

1:97567 

1:97610 

1:97653 

1:97695 

1:97737 

1:97779 

1:97820 

18 

72 

1:97820 

1:97861 

1:97901 

1:97942 

1:9 7 9 S1 

1:98020 

1:98059 

17 

73 

1:98059 

1:98098 

1:98136 

1:98173 

1:98210 

1:98247 

1:98284 

16 

74 

1:98284 

1:98320 

1:98355 

1:98391 

1:98425 

1:98460 

1:98494 

15 

75 

1:98494 

1:9S52S 

1:98561 

1:98594 

1:98626 

1:98658 

1:98690 

14 

76 

1:98690 

1:9S721 

1:98752 

1:98783 

1:98813 

1:98843 

1:98872 

13 

77 

1:98872 

1:9 S 9 01 

1:98930 

1:98958 

1:98986 

1:99013 

1:99040 

12 

73 

1:99040 

1:99067 

1:99093 

1:99119 

1:99144 

1:99169 

1:99194 

11 

79 

1:99194 

1:99219 

1:99243 

1:99266 

1:99289 

1:99312 

1:99335 

10 

80 

1:99335 

1:99357 

1:99378 

1:99400 

1:99421 

1:99441 

1:99462 

9 

81 

1:99462 

1:99481 

1:99501 

1:99520 

1:99539 

1:99557 

1:99575 

8 

82 

1:99575 

1:99592 

1:99610 

1:99626 

1:99643 

1:99659 

1:99675 

7 

83 

1:99675 

1:99690 

1:99705 

1:99719 

1:99734 

1:99748 

1:99761 

6 

84 

1:99761 

1:99774 

1:99787 

1:99799 

1:99811 

1:99823 

1:99S34 

5 

85 

1:99834 

1:99S45 

1:99855 

1:99865 

1:99875 

1:99885 

1:99894 

4 

86 

1:9 9 S 9 4 

1:99902 

1:99911 

1:99918 

1:99926 

1:99933 

1:99940 

3 

87 

1:99940 

1:99946 

1:99952 

1:99958 

1:99964 

1:99968 

1:99973 

2 

88 

1:99973 

1:99977 

1:99981 

1:999S5 

1:99988 

1:99991 

1:99993 

1 

89 

1:99993 

1:99995 

1:99997 

1:99998 

1:99999 

1:99999 

1:99999 

0 


60' 

50* 

40' 

30' 

20 ' 

10 ' 

0 ' 

Deg. 


Logarithm Cosine. 


Examples. Find the Logarithms, 

Log:sine 35° 40'? - - - - - = 1:76572, the answer. 

Log:cosine 18° 20' ? - - - - - = 1:97737, “ 


ti 





























62 


LOGARITHMS TANGENT. 


——- 

Dcs 

O' 

10' 

20' 

30' 

40' 

50' 

60' 

Deg 

0 

3:00000 

3:46372 

3:76476 

3:94085 

2:06580 

2:16272 

2:24192 

S9 

1 

2:24192 

2:30888 

2:36689 

2:41806 

2:46384 

2:50526 

2:54308 

88 

2 

2:54308 

2:57787 

2:61009 

2:64009 

2:66816 

2:69452 

2:71939 

87 

3 

2:71939 

2:74292 

2:76524 

2:78648 

2:80674 

2:82610 

2:84464 

86 

4 

2:84464 

2:86243 

2:87952 

2:89598 

2:91184 

2:92715 

2:94195 

85 

5 

2:94195 

2:95626 

2:91013 

2:98357 

2:99662 

1:00929 

1:02162 

84 

6 

1:02162 

1:03360 

1:04528 

1:05665 

1:06775 

1:07857 

1:08914 

S3 

7 

1:08914 

1:09940 

1:10955 

1:11942 

1:12908 

1:13854 

1:14780 

82 

8 

1:14780 

1:15687 

1:16577 

1:17449 

1:18305 

1:19146 

1:19971 

81 

9 

1:19971 

1:20781 

1:21578 

1:22360 

1:23130 

1:23887 

1:24631 

SO 

10 

1:24631 

1:25364 

1:26086 

1:26796 

1:27496 

1:28185 

1:28863 

79 

11 

l:28S65 

1:29534 

1:30195 

1:30846 

1:31488 

1:32122 

1:32747 

78 

12 

1:32747 

1:33364 

1:33973 

1:34575 

1:35169 

1:35756 

1:36336 

77 

13 

1:36336 

1:36909 

1:37475 

1:38035 

1:38588 

1:39136 

1:39677 

76 

14 

1:39677 

1:40212 

1:40741 

1:41265 

1:41784 

1:42297 

1:42805 

75 

15 

1:42805 

1:43308 

1:43805 

1:44298 

1:44787 

1:45270 

1:45749 

74 

1G 

1:45749 

1:46224 

1:46694 

1:47160 

1:47622 

1:48080 

1:48533 

73 

17 

1:48533 

1:48983 

1:49429 

1:49872 

1:50310 

1:50746 

1:51177 

72 

18 

1:51177 

1:51605 

1:52030 

1:52452 

1:52870 

1:53285 

1:53697 

71 

19 

1:53697 

1:54106 

1:54511 

1:54914 

1:55314 

1:55712 

1:56106 

70 

20 

1:56106 

1:56498 

1:56S8 7 

1:57273 

1:57657 

1:58038 

1:5S417 

69 

21 

L:5S417 

1:58794 

1:59168 

1:59539 

1:59909 

1:60276 

1:60641 

68 

22 

1:60641 

1:61003 

1:61364 

1:61722 

1:62078 

1:62433 

1:62785 

67 

23 

1:62785 

1:63135 

1:63483 

1:63830 

1:64174 

1:64517 

1:64858 

66 

24 

1:64858 

1:65197 

1:65534 

1-65870 

1:66204 

1:66536 

1:66867 

65 

25 

L66S67 

1:67196 

1:67523 

1:67849 

1:6S174 

1:68496 

l:6S81S 

64 

26 

1:68818 

1:69138 

1:69456 

1:69773 

1:70089 

1:70403 

1:70716 

63 

27 

1:70716 

1:71028 

1:71338 

1:71647 

1:71955 

1:72262 

1:72567 

62 

28 

1:72567 

1:72871 

1:73174 

1:73476 

1:73777 

1:74076 

1:74375 

61 

29 

1:74374 

1:74672 

1:74968 

1-75264 

1:75558 

1:75851 

1:76143 

60 

30 

1:76143 

1:76435 

1:76725 

1:77014 

1:77303 

1:77590 

1:7 7 S77 

59 

31 

1:77877 

1:78163 

1:78447 

1:78731 

1:79015 

1:79293 

1:79578 

58 

32 

1:79578 

1:79859 

1:80139! 1:80418 

1:80697 

1:80974 

1:81251 

57 

33 

1:81251 

1:81527 

1:81803 

1:82078 

1:82352 

1:82625 

1:82898 

56 

34 

L:S2S9S 

1:83170 

1:83442 

1:83713 

1:83983 

1:84253 

1:84522 

55 

35 

1:84522 

1:84791 

1:85059 

1:85326 

1:85593 

1:85860 

1:86126 

54 

36 

1:86126 

1:86391 

1:86656 

1:86920 

1:87184 

1:87448 

1:87711 

53 

O “7 

O i 

1:87711 

1:87974 

1:88236 

1:88498 

1:88759 

1:89020 

1:89281 

52 

38 

1:89281 

1:89541 

1:89801 

1:90060 

1:90319 

1:90578 

1:90836 

51 

39 

1:90836 

1.91095 

1:91352 

1:91610 

1:91867 

1:92121 

1:92381 

50 

40 

1:92381 

1:92637 

1:92894 

1:93149 

1:93405 

1:93661 

1:93916 

49 

41 

1:93916 

1:94171 

1:94497 

1:94680 

1:94935 

1:95189 

1:95442 

48 

42 

1:95443 

1:95697 

1:95926 

1:96205 

1:96458 

1:96712 

1:96965 

47 

43 

1:96965 

1:97218 

1:97471 

1:97725 

1:97978 

1:98230 

1:98488 

46 

44 

1:98485 

1:98736 

1:98989 

1:99242 

1:99494 

1:99747 

1:00000 

45 

Deg. 

60' 

50' 

40' 

30' 

20' 

10' 

0' 

Pee- | 


Logarithm Cotangent. 


The negative index is noted by two points, and must always follow an oppo¬ 
site operation to that of the mantissa. If the mantissa is added, subtract 
the index, and vice versa. 





































Logarithms Tangent. G3 


Deg 

0 ' 

10 ' 

20 ' 

30' 

40' 

50' 

CO' 

-. 

45 

0-00000 

0-00252 

0-00505 

0-00758 

0-01010 

0-01263 

o-oi51e 

44 

40 

0-01516 

0 01769 

0-02022 

0-02275 

0-02528 

0-02781 

0-03034 

43 

47 

0-03034 

0-03287 

0-03541 

0-03794 

0-04048 

0-04302 

0-04554 

42 

48 

0-04556 

0-04810 

0-05064 

0-05319 

0-05573 

0-05828 

0-0608; 

41 

49 

0-06083 

0-06339 

0-06594 

0-06850 

0-07106 

0-07362 

0-0761 

40 

50 

0-07618 

0-07875 

0-08132 

0-08389 

0-08647 

0-08904 

0-0916; 

39 

51 

0-09163 

0-09421 

0-09680 

0-09939 

0-10199 

0-10458 

0-10711 

38 

52 

0-10719 

0-10979 

0-11240 

0-11502 

0-11763 

0-12025 

0-12281 

37 

53 

0-12288 

0-12551 

0-12815 

0-13079 

0-13343 

0-13608 

0-13871 

36 

54 

0-13873 

0-14139 

0-14406 

0-14673 

0-14940 

0-15208 

0-15477 

35 

GO 

0-15477 

0-15746 

0-16016 

0-162S6 

0-16557 

0-16829 

0-17101 

34 

50 

0-17101 

0-17374 

0-17647 

0-17921 

0-18196 

0-18472 

0-18741 

33 

57 

0-18748 

0-19025 

0-19302 

0-19581 

0-19860 

0-20140 

0-20421 

32 

58 

0-20421 

0-20702 

0-20984 

0-21268 

0-21552 

0-21836 

0-22122 

31 

59 

0-22122 

0-22409 

0-22696 

0-22985 

0-23274 

0-23564 

0-23856 

30 

GO 

0-23856 

0-24148 

0-24441 

0-24735 

0-25031 

0-25327 

0*25624 

29 

61 

0-25624 

0-25923 

0-26222 

0-26523 

0 26825 

0-27128 

0-27432 

28 

62 

0-27432 

0-27737 

0*28044 

0-28352 

0-28661 

0-28971 

0-29283 

27 

63 

0-29283 

0-29596 

0-29910 

0-30226 

0-30543 

0-30861 

0-31181 

26 

6 4 

0-31181 

0-31503 

0-31826 

0-32150 

0-32476 

0-32803 

0-33132 

25 

65 

0-33132 

0-33463 

0-33795 

0-34129 

0-34465 

0-34802 

0-35141 

24 

66 

0-35141 

0-35482 

0*35825 

0-36169 

0-36516 

0-36S64 

0-37214 

23 

67 

0-37214 

0-37567 

0-37921 

0-38277 

0-38635 

0-38996 

0-39359 

22 

68 

0-39359 

0-39723 

0-40090 

0-40460 

0*40831 

0-41205 

0-41582 

21 

69 

0-41582 

0-41961 

0-42342 

0-42726 

0-43112 

0-43501 

0-43891 

20 

70 

0-43893 

0-44287 

0-44685 

0-45085 

0-45488 

0-45893 

0-46302 

19 

71 

0-46302 

0-46714 

0-47129 

0-47548 

0-47969 

0-48394 

0-48822 

18 

72 

0-48822 

0-49254 

0-49689 

0-50127 

0-50570 

0-51016 

0-51466 

17 

73 

0-51466 

0-51919 

0-52377 

0-52839 

0-53305 

0-53775 

0-54251 

16 

74 

0-54250 

0-54729 

0-55213 

0*55701 

0-56194 

0-56692 

0-57194 

15 

75 

0-57194 

0-57702 

0-58215 

0-58734 

0-59258 

0-59787 

0-60322 

14 

76 

0-60322 

0-60864 

0-61411 

0-61964 

0-62524 

0-63090 

0-63663 

13 

77 

0-63663 

0-64243 

0-64830 

0-65424 

0-66026 

0-66635 

0-67252 

12 

78 

0-67252 

0-67877 

0-6S511 

0-69153 

0-69804 

0-70465 

0-71134 

11 

79 

1-71134 

0-71814 

0-72503 

0-73203 

0-73913 

0-74635 

0-75361 

10 

80 

0-75368 

0-76112 

0-76869 

0-77639 

0-78422 

0-79218 

0-80021 

9 

81 

0-80028 

0-80853 

0-81694 

0-82550 

0-83422 

0-84312 

0-85211 

8 

82 

0-85219 

0-86145 

0-87091 

0-88057 

0-89044 

0-90053 

0-910S2 

7 

83 

1-91085 

0-92142 

0-93224 

0-94334 

0-95471 

0-96639 

0-97831 

6 

84 

1-97838 

0-99070 

1-00337 

1-01642 

0-02986 

1-04373 

1-05804 

5 

S5 

1-05804 

1-07284 

1-08815 

1-10401 

1-12047 

1-13756 

1-15531 

4 

80 

i -15535 

1-17389 

1-19325 

1-21351 

1 -23475 

1-25707 

1-28061 

3 

87 

1-28060 

1-30547 

1-33184 

1-35990 

1-38990 

1-42212 

1-45691 

2 

88 

1-45691 

1-49473 

1-53615 

1-58193 

1-63310 

1-69111 

1-75807 

1 

89 

1-75807 

1-83727 

1-93419 

2-05914 

1-23523 

2-53627 

0-00001 

0 


60' 

50' 

40' 

30' 

20 ' 

10 ' 

0 ' 

Deg 


Logarithm Cotangent. 


Example. Find the Logarithms, 

Lo'J-.tan. 38° 40'?.= 1:87185, the answer. 

Log.tan. 5S° 50' ?---**= 0:21836, 

























61 


Arithmetical Progression. 


ARITHMETICAL PROGRESSION. 

Arithmetical Progression is a series of numbers, as 2, 4, 6, 8, 10,12, 
&c., or 18, 15, 12, 9, 6, 3, in which every successive term is increased or dimin¬ 
ished by a constant number. 

Letters will denote, 
a — the first term of the series. 
b = any other term whose number from a is n. 
n — number of terms within a and b. 

= the difference between the terms. 

5 = the sum of all the terms. 

In the series, 2, 5, 8,11, a = 2, b = 11, n = 4, d == 3, and 5 — 26. 

4®^When the series is decreasing, take the first term = b and the last term 

= a. 

The accompanying Table contains all the formulas or questions in Arithmeti¬ 
cal Progressions, and the nature of the question w ill tell which formula is to be 
used. 

Formulas for Arithmetical Progressions. 


a = 

Cr 1 

1 

1 

M 

• 

1 

* 1, 

S 

6 —a 
= £=l’ 

* 9, 

a — 

25 

— — &, 

- 2, 

S 

1 

It 

rC> 

'—«*’ 

II 

- io, 

a = 

5 * 

n 2 

- 3, 

s 

• 

J i 

*2, ^ 

II 

* 11, 

b = 

a-J-c> (7? —1), 

- 4, 

s 

2 (bn—S)_ 
n (7i—1) 

• 12, 

b = 

25 

-a, - 

n 

- 5, 

s 

7i (a-WO 

-- , - - 

- 13, 

,b = 

-P-yn-1), - 

71 2 

- 6, 

5 

(a-\b)(b-\-n — a). 

21 

- 11, 

n = 

- 

- 7, 

5 

= n [o+f (n—1) 

15, 

n = 

25 

a+6’ 

- 8, 

5 = 

= n[b— ^(n-1)] - 

- 16, 


— 1 ±>/(^+])- 25 « 


b = - 
2 


+2 ?5, 


n 


± \/R? 

4-;-± v /(H)+x.- 


1 b 

n = — -f- 


3 , h , f \ by 25 

ruy U+sJ-"-’ 


17, 

18, 

19, 

20 , 






















r 


Arithmetical Progrsssiox. 


65 


Example 1. A man was engaged to dig a well at one dollar ($1) for the first 
foot ot the depth ot the well, $PS4 for the second, and 84 cents more per every 
successive toot in depth, until he reached the water, which was found at a depth 
■>i 25 feet. How much money is due to the man? 

This will be answered by the formula 15, in which a = l, <2 = 0-84, and 
n = 25, then the sum, 

P 0 84 "1 

S — 25 1 + — 7 ,- (25 - 1)J = $277 the answer. 

Example 2. A Propeller ship which is to run between Philadelphia and 
Charleston, cost $116500, of which the company agreed to pay cn account 
$14075 at her first trip to Charleston; and per every successive trip, they paid 
$650 less than the former. How many trips must the vessel make until she is 
fully paid ? 

This will be answered by the formula 20, in which b = $14075, d = 650, and 
S = 116500. 


„ _ I_i_ 14 °7 5 _. / /14075 . 1 \ a 2X116500 , 

n - 2 + C60 \/ l £50 + 2 ) - mi T- = 10 ' 6 or 11 tnps - 

Arithmetical Progressions of a Higher Order. 

Arithmetical Progressions are of the first order, when the difference S is a 
constant number, but when the difference $ progresses itself with a constant 
number, the Progression is of the second order. 

When the difference £ progresses in a second order, the Progression is of the 
third order, Ac., &c., and is thus explained: 

, . . n, - - Arith. Prog., first order. 

n (n-j-1) 


1, 2, 3, 4, 5, 6 , . 

1, 3, 6 , 10, 15, 21, 


1, 4, 10, 20, 35, 56, 


1, 5, 15, 35, 70, 126, 


n (n+l)(n+ 2 ) 
2X3 ’ 

n (re+l')(r?-+2)(7i+3) 

2X3X4 * 


2 d. order. 


3d. order. 


4th. order. 


Here you will discover that the sum of n terms in one order, is equal to the 
same nth term iu the next higher order. Arithmetical Progressions of the first, 
second, and third orders, are applied to 

PILES OF BALLS AND SHELLS. 

Triangular Piling. 

Example 1. A complete triangular pile of balls has n = 12 balls in each side. 
Require how many balls in the base, and how many in the whole pile ? 

12 r 2+1) 

2 


In the base, 
Whole pile, - 

1, 4, 9, 16, 25, 36, • 
3, 5, 14, 30, 55, 91,. 


= 73 balls, 


- 2 d. order. 


^ 2 ( 12 + 1 ) 22 + 2 ) . 

2X3 ’ 

Square Piling. 

. . n 2 

n(n+\)(2n+l) 

2>o * 


3d. order. 

2 d. order. 
3d. order. 


[Nee Examples 2 and 3 on page 67.] 


t> * 











66 


Geometrical Progression. 


GEOMETRICAL PROGRESSION. 

Geometrical Progression is a series of numbers, as 2: 4 : S : 16:32 : Ac., 
or 729:243: 81: 27 :9 : Ac., in which every successive term is multiplied or divided 
by a constant factor. 

Letters will denote , 

a — the first term of the series. 

b — any other term whose number from a is n. 

n = number of terms within a and b. 

r = ratio, or the factor by which the terms are multiplied or divided. 

S = Sum of the terms. 

In the series 1 : 3 : 9 : 27 : a = 1, b = 27, n — 4, r — 3, S = 40. 

The accompanying Table contains all the formulas or questions in Geometrical 
Progressions. The nature of the question will tell which formula is to be 
used. 


Fornmlas for Geometrical Progressions. 


b 

a — -, 

rn—Y 

- 

1 , 

•‘- 1 nr 

1 ~ \ / a * 

- 7, 

a =- S—r (S- 

-to,- - 

2, 

-5 

II 

On On 

1 1 

ft 

i 

• 

i 

8, 

a 

11 

On 

u 

- 

3, 

ar n -\-S—rS — a = 0, 

- 9, 

b = ar n ~*, 

- 

4, 

1 

1 

1 

e r—i 

l i 

•O ** 

1 

sq 

10 , 

b-s s ~ a - 


ft 

o aO* 71 — 1) 

- U, 

r 

? 


* 1 - r ’ * * 



6 , 

e b (r n — 1) 

(?• — l)r«-i ’ 

12 , 


n - l + l2E±-I^ a , - - 

log.r 

- - - - 13, 

log.b — log.a 

” ~ 'log.(S-\-a) — log.(S — bf 

.... 14, 

log,[a+S(.r — 1)1 — Jng.a 
log.r. 5 

- - - - 15, 

^ , log.b — tog.fhr <S7r—1)] 

log.r. * 

- - - - 16, 

, n—1 — ft — I'-’ 

o Vo — a v a 

O —-. — 1 --- y 

n —1/ . n ~.\r— 

v d — v a 

- - - i7, 


























0EGMHTIUCAL PROGRESSION. 


67 


Example!. Required the 10th term in the Geometrical Progression 4:12:36.... ? 
Given a 4, n = 10, and r — 3. We have, 

Formula 4. b — ar 71 — 1 = 4X3 9 = 78732, the tenth term. 

Example 2. Required the sum of the 10 terms in the preceding example ? 

4(310 _i) 


Formula 11, 


s _ °-t T - 1 ) 

r — 1 


118096, the sum. 


Example 3. Insert 6 proportional terms between 3 and 384 ? 
Given a = 3, b — 384, and n — 6+2 = 8. 

«—l 


Formula 7, 


then 


r — 


V 4 + 


384 


' — 2 , 


3 : 6 :12 : 24 : 48 : 96 :192 : 384, the answer. 


Example 4. A man had 16 twenty dollar gold pieces, which he agreed to ex¬ 
change for copper in such a way, that he gets one cent on the first $20, two on 
the second, four on the third, and eight on the fourth, &c., &c.; until the sixteen 
$20 pieces were covered. IIow many cents will come on the sixteenth gold 
piece, and what will be the whole amount of copper on the gold ? 

In the progression 1 : 2 : 4 : 8 : &c., we have, 

Given n = 16, r =2, and a — 1, then, 


Formula 4. 


1X2 I6— * = — = 9-= ^ ~ = 32768 cents, on the 


216 4 8 164 2563 


2 2 

sixteenth piece. 

The total sum of cents will be found by the 


Formula 10. 


S = 


32768X2—1 


- = 65535 cents = $655-35. 


Piling of Balls and Sheds.—[From page 64.] 

i Example 2. IIow many balls are contained in a complete square pile, n — 10 
rows? 

io(io+i)(2xio+i) = iox nx2 i _ 385 

2X3 6 

Rectangular Filing. 

Let m be the number of balls on the top of the complete pile, and n — num¬ 
ber of rows in the same, then the number of balls in the whole pile will 
be, 

n(«+l)f2.»+3«-2) . . . 

2X3 * 

The number of balls in the longest bottom side will be = m+n — 1. 

Example 3. The rectangular pile having 15 rows and 23 balls on the the top, 
how many in the whole pile? 




15(15+1)(2X15+3X23—2) 15X16X67 

2X3 6 


= 2680 balls. 




















Compound Interest.—Annuities. 


«S 


COMPOUND INTEREST. 

Compound Interest is when the Interest is added to the Capital for 
each year, and the sum is the Capital for the following year. 

« = c( 1+P) n , 


Amount, 

Capital, 

Per centage, 

Number of years, 


c = 


(1+P)*’ 

”/«" -r 

P — V - —I, 


n = l°Q-a — %-c , 

log. (i-bp) 


1. 

* 2 , 
S, 

- 4, 


>g£g?-In these formulas p must he expressed in a fraction of 100. 

Example 1. A capital c = 8050 standing with Compound Interest at p = 5 
per cent, what will it amount to in n — 9 years. 

A m&iint a = S050 (405)9 = 13419 dollars. 

Example 2. A man commenced business with c — 300 dollars, after n = 5 
years he had a = 0875 dollars At what rate did his money increase, and 
how soon will he have a fortune of 50000 dollars? 

The first question, or the pgr centage, will be answered by the formula 3. 


P = v' ~ —1 = V 22*9166 — 1 = 0-87, or 87 per cent. 

oUO 

The time from the commencement of business until the fortune is completed, 
will be answered from the formula 4. 

fogr.50000-— 7«7.800 4’69S97—247712 _ 

“ “ l,, g . 1 - w - " —0^27100®— “ 8 ' 16! ’ ) ' CarS ' 

or S years and 2 months. 




ANNUITIES. 

A minify is a certain sum of money to he paid at regular intervals. 

A yearly payment or annuity b, is standing for n years, to find the whole 
amount a at p per cent. Interest. 


Amount, 


Amount, 


a — bn (n-fl)J Simple Int„ 

a = — [(l-hp) n —l] Comp. Int., - 


1, 


. o 


A yearly payment or annuity b , is to he paid for n years, to find the present 
worth, or the amount a, which would pay it in full, at the beginning of the 
time n, deducting^ per cent. Interest. 


Amount, 
Amount, 


a-S»[l-|(^±f)]simp. lDt -’ 

a = — f" 1 — - 1 . 1 Comp. Int., 

p L (i+jO nJ 


3, 

4, 





























Annuities.—Paper.—Selection of Water Colours. 


69 


A debt D, standing for Interest, is diminished yearly by a sum b ; to find the 
debt d after n years, and the time n when it is fully paid? 

The debt d after n years will be, 


d 




- 5 , 


n = 


6 . 


The time n until fully paid will be, 

l og.b — log.(I? Dp) 

%¥if) 

If 6 = Dp then n = GO, or the debt D will never be paid. If b<Dp, the debt 
D will be increased. 

To find the yearly annuity b, which will pay a debt D in n years, at p per 
cent. Compound Interest? 


b 


Dp 


7 . 


PAPER. 

1 Ream = 20 quires = 480 sheets. 

1 quire = 24 sheets. 
Drawing Paper. 


Cap, 

Demy, 

Medium, 

Royal, 

Super Royal, - 
Imperial, - 
Elephant, 


13X16 inches. 
20X15 “ 

22X17 “ 

24X19 “ 

27X19 “ 

30X21 “ 

28X22 ££ 


Columbier, 

Atlas, 

Theorem, - 
Double Elephant, 
Antiquarian, - 
Emperor, - 
Uncle Sam, - 


34X23 inches. 
33X26 ££ 

34X28 ££ 

40X26 ££ 

52X31 “ 

40 X 60 ££ 

48X120 ££ 


Glazed or Crystal, 
Yellow or Blue Wove. 


Continuous Colossal Drawing Paper, No. A, and No. B, 56 inches wide, and of 
any required length. No. A, of this paper is excellent for mechanical drawings. 
Price from 40 to 50 cents per yard. 

Tracing Paper. 

Double Crown, 30 by 20 inches. 

Double Double Crown, 40 ££ 30 ££ 

Double Double Double Crown, 60 ££ 40 ££ 

Finest French Vegetable Tracing Paper. 

Grand Raisin (or Royal) 24 in. by 18. Grand Aigle 40 in. by 27. - 

Mounted Tracing Paper. 

This paper is mounted on cloth, and is still transparent; it will take ink and 
water colours. It is 38 inches wide, and of any required length. 

Vellum Writing Cloth, 

Adapted for every description of tracing; it is transparent,durable, and strong. 
It is 18 to 38 inches wide, and of any required length. 

-- 


SELECTION OF WATER COLOURS 


Blue. 

it 

Real Ultramarine. 

French Blue. 

Red. Rose Madder. 

<£ Light Red. 

cc 

Indigo. 

Brown. Vandyke. 

it 

Cobalt Blue. 

Brown Madder 

Green. 

Olive Green. 

Black. India Ink. 

Yellow. 

Cadmium. 

££ Blue Black. 

it 

Gamboge. 

“ Ivory Black. 

it 

Ochre. 

££ Lamp Black. 

Red. Carmine. 

White. Chinese White. 


££ Crimson Lake. 
















70 United States’ Standard MeaSDhes and Weights. 



MEASURE OF LENGTH. 

The Standard Measure of Length is a brass rod = 1 yard at the temperature of 
32° Fahrenheit. The length of a pendulum vibrating seconds in vacuo, at 
Philadelphia is 1*08614 yards, at + 32° Fahrenheit. 

The Surveying Chain is = 22 yards — CG feet. It consists of 100 links, 
and each link = 7*a2 inches. 

ROPES AND CABLES. 

1 Cable length = 120 fathoms = 720 feet. 

1 fathom = G feet. 

GEOGRAPHICAL AND NAUTICAL MEASURES. 

1 Degree of the great circle of the Earth round the Equator = 69*545 statute 
miles = €0 Nautical miles. 

1 Statute mile = 5280 feet = 0*S6275 Nautical miles. 

1 Nautical mile = 6120 feet = 1*159 Statute miles. 

LOG LINE. 

The I.og Liinc should be about 150 fathoms long, and 10 fathoms from 
the Log to the first knot on the line. If half a minute glass is used, it will be 
51 feet between each succeeding knot. For 28 seconds glass it will be 47*6 feet 
= 7*93 fathoms per knot. This is the length of knot by calculation, but prac¬ 
tically it is shortened to 7*5 fathoms per knot for 28 seconds glass. 

MEASURE OF CAPACITY. 

Gallon* The standard Gallon measures 231 cubic inches, and contains 
8-3388S22 pounds Avoirdupois = 583721757 grains Troy, of distilled water, at its 
maximum density 39*S3° Fahrenheit, and 30 inches barometer height. 

BusIieS. The standard Bushel measures 2150*42 cubic inches = 77*627413 
pounds Avoirdupois of distilled water at 39*83° Fahrenheit, barometer 30 inches. 
Its dimensions are 18^ inches inside diameter, 19£ inches outside, and 8 inches 
deep; and when heaped, the cone must not be less than 6 inches high, equal 
2747*70 cubic inches for a true cone. 

Pound* The standard Pound Avoirdupois is the weight of 27*7015 cubic 
inches of distilled water, at 39*83° Fahrenheit, barometer 30 inohes, and 
weighed in the air. 


MEASURE OF LENGTH. 


Miles. 

Furlongs. 

Chains. 

Rods. 

Yards. 

Feet. 

Inches. 

1 

8 

SO 

320 

1760 

5280 

03360 

i 0125 

1 

10 

40 

2*~0 

C60 

7920 

00125 

01 

1 

4 

22 

CG 

792 

0003125 

0 025 

025 

1 

5.5 

16*5 

198 

0 00055818 

00045454 

0*045454 

0*]?1S18 

1 

3 

36 

000018939 

0 00151515 

0 01515151 

0*0 (U) 0 

0*33333 

1 

12 

'0 000.015783 i0*0001262t>2|0*00126262l 

0*00505050 |0*0277777 

0*083333 

1 


MEASURE OF SURFACE. 


Sq. Miles. 

Acres 

S.Chains. 

Sq. Rods. 

Sq. Yards. 

Sq. Feet. 

Sq. Inches. 

1 

CIO 

C400 

1024CO 

3037000 

27878400 

4014489000 

0 001562 

1 

10 

110 

4840 

435(0 

C9C9G0 

0 00015 2 

0*1 

1 

16 

484 

4356 

G3G90 

0090009764 

0 00625 

0 0 25 

1 

30*25 

272 25 

39204 

0 000000323 

0 0002058 

0 002066 

0*0330 

1 

9 

1296 

0 0000000358 

0 00()09229( |0*000u229C 0 003 7 

0*1111111 

1 

144 

0*00000000025 

0 00C0031-1S 

0*0000014310*00002552 20 000771C 

0006944 

1 

t 
































United States’ Standard Measures' and Weights. 71 









.... 

- 




MEASURE OF CAPACITY. 



Cub. Yard. 

Barrels. 

Bushels. 


Cub. Feet. 

Pecks. 

Gallons. 

Cub. Inch 

1 

50103 

25-2467 


27 

1CO-087 

£01-974 

46056 

01782 

1 


45 


4-0125 

18 

30 

8310 

0 03911 

9'2222 


1 


1-2438 

4 

8 

215042 

0-037037 

3 2078 


0*804 


1 

3-73800 

7-.47C19 

1728 

0 00 02 

0*05555 


0-25 


0-20738 

1 

o 

412 

IrVllH <JJ 

tj / / / 


0 125 


0-13369 

0 o 

i 

B31 

0 00002113 

1) 0001202 

0000405 


00005787 

00021045 

0 004320 

i 

! 

< 

MEASURE OF LIQUIDS. 

! 

Gallon. 

Quarts. 


Pints. 

Gills. 

Cub. inch. 


1 


4 


8 

32 

231 


02.5 



1 


2 

8 

57-75 


0 12.5 

0-5 


i 

4 

2^-87; 

-J 

003125 

0-125 

0 25 

1 

7-2175 

0*004329 

--- - -.- 

0 017315 

-- 

0 03403 

0-13858 

1 

t 


MEASURES OF WEIGHTS. 


AT OIRDUPOIS. 


Ton. 

Cwt. 

Pounds. 

1 

Ounces. . 

-:-- 

Drams. 

1 

20 

2240 

35840 

573440 

0 05 

1 

112 

1792 

28072 

0-00044642 

0 0081285 

1 

hi 

256 

0 00002710 

0-000558 

00625 

i 

16 

0 00000174 

0 0000348 

0-0016 

0-0 25 

1 

TROY. 

Pounds. 

Ounces. 

Dwt. 

Grains. 

Pound Avoir. 

i 

12 

240 

5760 

0822861 

0-083333 

1 

20 

480 

0 0H8 573 

0 004166 

0-05000 

1 

24 

0 60342-5 

0-0001736 

0-00208333 

0-0416666 

1 

0 C0020571 

I 215275 

1458333 

29-1666,6 

4861 

1 


APOTHECARIES’. 


Pounds. 

Ounces. 

Drams. 

Scruples. 

Grains 

* 

12 

96 

288 

5760 

0-08333 

1 

8 

24 

480 

0-01041666 

0 125 

1 

3 

60 

0 0034722 

0 0416666 

0-3333 

1 

20 

0 00017361 

0020833 

0-16666 

§ 

1 

























































72 


United States’ Standard Measures and Weights. 


DIAMOND. 


Carat. 

Grain. 

Parts. 

Grains. Troy. 

1 

4 

04 

32 

025 

1 

10 

0-8 

0 015625 

00025 

1 

005 

0-3125 

1-25 

20 

1 


GOLD COINS. U. S. STANDARD WEIGIIT. 

WEIGHT TEOY. 



Tv -i -I 


Ounces. 

HU16 01 tll6 L01I13 

Dollars. 

Grains. 

Double Eagle ...... 

$ 20 

516 

1075 

Eagle 

$ 10 

258 

0 5375 

Half Eagle . 

$ 5 

129 

0 26875 

Three Dollar piece .... 

$ 3 

77*4 

0 16125 

Quarter Eagle . . . . . 

$ 2-50 

64 5 

0134375 

Dollar piece . 

$ 1 

25 8 

005375 

Value per Grain . 

$ 00387500 

1 

0 00208333 

Value per Ounce ..... 

$ 18-C046 . 

480 

1 


SILVER COINS. U. S. STANDARD WEIGIIT. 




WEIGHT TROY. 

Name of the Coins. 

Cents. 

Grains. 

Ounces. 

One Dollar . ..... 

100 

3F4 

0-8 

Half Dollar or five Dimes . . . 

50 

192 

04 

Quarter Dollar or 2| Dimes 

25 

06 

0-2 

One Dime. 

10 

38-4 

0 08 

Half Dime. 

5 

19-2 

0 04 

Three Cents piece . . . • . 

3 

1152 

0024 

Value per Grain. . ..... 

0-26043668 

1 

0 00208333 

Value .per Ounce. 

125 

480 

1 

Copper Cent . . 

1 

168 

0 35 

Half Cent. 

05 

84 

0 175 

Value per Grain . . . . . 

000595238 

1 

0 00208333 

Value per Ounce ..... 

2-8571424 

480 

1 

{ The Standard fineness of Gold and Silver Coins 

is one weight of alloy to 


nine weights of pure metal. The alloy for Gold Coin is Silver and 
Copper, and Copper for Silver Coin. 


Relative value of Foreign Gold and Silver Coins, fixed by the law of the 

United States. 


1 Pound Sterling of Great Britain. . $ 4*84 

1 Shilling . 0242 

1 Pound Sterling of Nova Scotia, New Brunswick, Newfoundland and 

Canada. .400 

1 Dollar of Mexico, Peru, and Central America.. . 100 

1 Pagoda of India .. . ... . . . L84 

1 Real Vellon of Spain.0 05 

1 Real Plate of Spain.•.0-10 

1 Rupee Company ..0 441 

1 Rupee of British India. ..0-44? 

J F>-anc of France and Belgium ^ . 0-18^ (V 

1 Specie Dollar of Sweden and Norway.100 

1 Ducat of Sweden ... 2' 15 

1 Specie Dollar of Denmark. 1-05 

1 Florin of Netherland.0.40 . 

1 Florin of Southern States of Germany.• . . . 0.40 

1 Guilder of Netherland.0.40 

1 Livre Tournoise of France.. 0.18£ 






























































Foreign Weights and Measures. 


73 


1 Livre of the Lombardy Venitian Kingdom . . . 

1 Livre of Tuscany.. . , 

1 Livre of Sardinia .... . 

, 1 MUrea of Portugal . . 

1 Mllrea of Azores . 

1 Marc Banco of Hamburg. 

1 Rix Dollar or Thaler of Prussia and the Northern States of Germany, 

1 Rix Dollar of Bremen.. 

1 Rouble Silver of Russia. 

1 Pterin of Austria.. 

1 Ducat of Naples. 

1 Ounce of Sicily. 

1 Tad of China.•. 

1 Livre of Leghorn.•, 


$0.16 | 
0.16 
O.lS^o 
1.12 
0.83$ 
0-35 
0.69 
0.7Sf 
0.75 
0.48| 

o.so 

2.40 

1.43 

0.16 


FOREIGN MEASURES OF LENGTH COMPARED WITH AMERICAN. 


Places. 

Measures. 

Inches. 

Places. 

Measures. 

Inches. 

Amsterdam 

Foot 

11-14 

Malta . 

Foot 

11-17 

Antwerp 

%i 

11-24 

Moscow 

Ci 

13-17 

Bavaria . 

a 

11-42 

Naples . 

Palmo 

10-38 

Berlin . 

a 

12-19 

Prussia 

Foot 

12-36 

Bremen . 

cc 

11-38 

Persia . 

A rish 

38-27 

Brussels 

a 

11-45 

Rhineland 

Foot 

12-35 

China 

“ Mathematic 

13-12 

Riga . . 

a 

10-79 

i. 

“ Builder’s 

12*71 

Rome . 

a 

11-60 

it 

“ Tradesman’s 

13-32 

Russia . 

a 

13-75 

a 

“ Surveyor’s 

12-58 

Sardinia . 

Palmo 

9 78 

Copenhagen 

(( 

12-35 

Sicily . 

it. 

9-53 

Dresden 

U 

11-14 

Spain . 

Foot 

n -03 

England 

a 

12-00 

U 

Toesas 

66'72 

Florence 

Braccio 

21-60 

a 

Palmo 

8-34 

France . 

Pied de Roi 

12-79 

Strasburgh 

Foot 

11-39 

.< 

Metre 

39-381 

Sweden 

i. 

11-69 

Geneva 

Foot 

19-20 

Turin 

a 

12-72 

Genoa . 

Palmo 

9-72 

Venice . 

u 

13-40 

Hamburgh . 

Foot 

11-29 

Vienna 

a 

12-45 

Hanover 

a 

11-45 

Zurich • 


11-81 

Leipsic . 

n 

11-11 

Utrecht 

a 

10-74 

Lisbon . 

a 

12-96 

W arsaw 


14-03 

a 

Palmo 

8-64 

... 


. _ I 


ENGLISH AND FRENCH MEASURES OF LENGTH. 


British. 


French. 


Yard is referred to a natural standard, which is the length of a pendu¬ 
lum vibrating seconds in vacuo iu London, at the level of the sea; 
measured on a brass rod, at the temperature of 62° Fahrenheit, 
= 390393 Imperial inches. 


Old System- 

-1 Line = 12 points . 

= 0-08884 U. S. inches. 

1 Inch = 12 lines . 

= 1 06604 “ 


1 Foot = 12 inches . 

12-7925 “ 


1 Toise = 6 feet 

'== 76-755 “ 


1 League = 2280 toises . 

^common.) 


1 League — 2000 toises . 

1 Fathom = 5 feet. 

(post.) 

New System.- 

-1 Millimetre 

= -03939 U. S. inches. 

1 Centimetre • " • 

=r -39380 “ 


1 Decimetre • 

= 3-93809 “ 


1 Metre . 

= 39-38091 “ 


1 Decametre . • • 

= 393-80917 “ • 


1 Hecatometre 

= 3933-09171 “ 


7 













































?4 Foreign Weights and Measures. 


FOREIGN ROAD MEASURES COMPARED WITH AMERICAN. 

Places. 

Measures 

Yards. 

Places. 

Measures. 

Yards. 

Arabia . * 

Mile 

2148 

Hungary . 

Mile 

9113 

Bohemia 

66 

10137 

Ireland 

66 

• 3038 

China . 

Li 

629 

Netherlands 

66 

1093 

| Denmark . 

Mile 

8244 

Persia . 

Parasang 

6086 

England 

“ Statute 

1760 

Poland 

Mile, long 

8101 

66 

“ Geographical 

2025 

Portugal . 

League 

6760 

Flanders 

66 

6869 

Prussia 

Mile 

8468 

France . 

League, marine 

6075 

Rome , 

66 

2025 

66 

“ common 

4861 

Russia . 

Verst 

1167 

a 

“ post 

4264 

Scotland . 

Mile 

1984 

Germany • 

Mile, long 

10126 

Spain . 

League, common 

7416 

Hamburgh . 

66 

8244 

Sweden 

Mile 

11700 

Hanover 

66 

11559 

Switzerland 

66 

9153 

Holland 

66 

6395 

Turkey 

Berri 

1826 

MEASURES OF SURFACE. 

French. Old System. — 1 Square Inch . . = 1T364 U. S. inches. 



1 Arpent (Paris) . . = 900 s 

1 Arpent (woodland) = 100 s 

quare toises. 
quare royal perches. 

New System .—1 Arc 

. 

. . ■= 100 square metres. 



1 Decare 

1 Hecatare . 

1 Square Metre 

• • 10 ares. 

. . = 100 ares. 

. • — 1550-85 square inches, 

or 

10-7698 square feet. 

1 Arc.= 1070-98 “ 

FOREIGN MEASURES OF SURFACE COMPARED WITH AMERICAN. 

Places. 

Measures. 

Sq. Yds. 

Places. 

Measures. 

Sq. Yds. 

Amsterdam 

Morgen 

9722 

Portugal . 

Geira 

0970 

Berlin . 

“ great 

6786 

Prussia 

Morgen 

3053 

66 

“ small 

3054 

Rome . 

Pezza 

3158 

Canary Isles 

Fanegada 

2422 

Russia . 

Dessetina 

13066-6 

England 

Acre 

4840 

Scotlahd . 

Acre 

(450 

Geneva 

Arpent 

6179 

Spain . 

Fanegada 

5500 

Hamburgh . 

Morgen’ 

66 

11545 

Sweden 

Tunneland 

5900 

Ilancver 

3100 

Switzerland 

Faux 

7855 

Ireland . 

Acre 

7840 

Vienna 

Joch 

6889 

Naples • 

Moggia 

3998 

Zurich 

Common acre 

3875-0' 


FOREIGN MEASURES OF CAPACITY. 

1 

■ | 

British. The Imperial gallon measures 

277-274 cubic, inches, containing 10 lbsv 

Avon dupois of distilled water, weighed in air. at the temperature 

of 62° degrees, the barometer at 30 inches. 

For Gh'ain. 8 bushels = 1 quarter. 

1 quarter = 10'2694 cubic feet. 

Coal, or heaped measure. 3 bushels — 1 rack. 


12 lacks — 1 chaldron. 

Imperial bushel — 2218T92 cubic inches, 

*Heaped bushel. 191 ins. diam.,cone 6 ins. high = 2812-4872 cubic ins. 

1 chaldron = 58-058 cubic feet, and weighs 3136 pounds. 


1 chaldron (Newcastle) = 5936 pounds. 



French. New System. — 1 Litre 

= 1 cub. decimetre, or 61-074 U. S. cubic inches. 

Old System. — 1 Poisseau = 13 litres = 793-964 cub. ins., or 3-43 galls. 

1 Pinte = 0-931 litres, Or 56-817 cubic inches. 
Spanish. 1 Wine Arrobn. = 4-2455 gallons. 

1 Fanega (common measure) = 1-593 bushels. 



* When heaped in the form of a true cone. 





























Foreign Weights and Measures. 


76 


FOREIGN LIQUID MEASURES COMPARED WITH AMERICAN. 


Places. 


Measures. 

Cub. In. 

Places. 


i Measures. 

Cub. In- 

! Amsterdam . 


Auker 

2331 

Naples . 


Wine Barille 

2544 

66 


Stoop 

146 

66 


Oil Stajo 

1133 

Antwerp 


66 

194 

Oporto . . 


Almude 

1555 

Bordeaux . 


Barrique 

14033 

Rome . 


Wine Barille 

2560 

j Bremen 


Stubgens 

194-5 

66 


Oil “ 

2240 

| Canaries . 


Arrobas 

949 

66 


Boccali 

80 

1 Constantinople 


Almud 

319 

Russia . . 


Weddras 

. 752 

Copenhagen . 


Anker 

2355 

66 


Kunkas 

94 

Florence . 


Oil Barille 

1946 

Scotland . 


Pint 

103-5 

66 


Wine “ 

2427 

Sicily 


Oil Caffiri 

662 

France 


Litre 

61-07 

Spain . 


Azumbres 

22-5 

Geneva 


Setier 

2760 

66 


Quartillos 

30-5 

Genoa 


Wine Barille 

4530 . 

Sweden 


Eimer 

4794 

66 


Pinte 

90-5 

66 


Kanna 

159-57 

Hamburgh 


Stubgen 

221 

Trieste 


Orne 

4007 

Hanover . 


66 

231 

Tripoli 


M attar i 

1376 

Hungary . 


Eimer 

4474 

Tunis . 


Oil ‘- 

1157 

Leghorn 


Oil Barille 

19421 

Venice 


Secchio 

628 

Lisbon . 


Almude 

1040 

Vienna 


Eimer 

3452 

Malta . 


Caffiri 

1270 

66 


Maas 

86-33 

FOREIGN DRY MEASURES COMPARED WITH AMERICAN. 

Places. 


Measures. 

Cub. In. 

Places. 


Measures. 

Cub. In. 

Alexandria . 

Rebele 

9587 

Malta . . 


Salme 

16930 

66 

Kislos 

10418 

Marseilles 


Charge 

9411 

Algiers . • 

Tarrie 

1219 

Milan . 


Moggi 

8444 

Amsterdam 

Mudde 

6596 

Naples 

• 

Tomoli 

3122 

66 

Sack 

4947 

Oporto 

• 

Alquiere 

1051 

Antwerp 

Yiertel 

4705 

Persia 

• 

Artaba 

4013 

Azores . 

Alquiere 

731 

Poland 


Zorzec 

3120 

Berlin . . 

Scheffel 

3180 

Riga . . 


Loop 

3078 

Bremen . 


66 

4339 

Rome . 


Rubbio 

16904 

Candia . 

Charge 

9288 

66 


Quarti 

4226 

Constantinople 

Kislos 

2023 

Rotterdam 


Sach 

6361 

Copenhagen 

Toende 

8489 

Russia 

• 

Ohetwert 

12448 

Corsica . 

Stajo 

6014 

Sardinia . 

• 

Starelli 

2988 

Florence 

Stari 

1449 

Scotland . 

• 

Firlot 

2197 

Geneva . 

Coupes 

4739 

Sicily . . 


Salme gros 

21014 

Genoa . 

Mina 

7382 

66 


(( generale 

16886 

Greece • 

Medimni 

2390 

Smyrna . 

• 

Kislos 

2141 

Hamburgh . 

Scheffel 

6426 

Spain . 


Catrize 

41269 

Hanover 

Mai ter 

6868 

Sweden . 


Tunna 

8940 

Leghorn . 

Stajo 

1501 

Trieste . 

• 

Stari 

4521 

66 

Sacco 

4503 

Tripoli 

• 

Caffiri 

19780 

Lisbon . . 

Alquiere 

817 

Tunis . 


66 

21855 

66 

Fanega 

3268 

Venice 


Stajo 

4945 

Madeira . . 

Alquiere 

684 

Vienna 


Metzen 

3753 

Malaga . . 

Fanaga 

3783 







FRENCH MEASURES OF SOLIDITY. 


French. 1 Cubic Foot 

• • 

• - 

= 2093-470 U. S. inches. 

Decistre 

. • 

• • 

• • =3 

3-5375 cubic feet. 

Stere (a cubic metre) 

• • 

. . s 


25-375 “ 


Decastere . 

• • 

• • s 

= 353-75 « 


1 Stere 


• • 

• • 

. . = 

= 61074"664 cubic inches. 

For the Square and Cubic Measures of other countries, take the length of the 

measure in Table, 

page 72, and square or cube it as required. 







































76 


Foreign Weights and Measures, 


1 


ENGLISH AND FRENCH MEASURES OF WEIGHT. 

British. 1 troy Grain = -003961 cubic inches of distilled water. 

1 troy Pound = 22-815689 cubic inches of water. 

French. Old System. —1 Grain 


1 Gros 
1 Ounce 
1 Livre 

New System. —Milligramme 
Centigramme 
Decigramme 
Gramme 
Decag'ramme 
Hecatogramm e 


1 Kilogramme 
1 Pound avoirdupois 
1 Pound troy 


0-8188 grains troy. 
58-9548 “ 

1-0780 oz. avoirdupois. 
1-0780 lbs. “ 

•01543 troy grains. 
-15433 “ 

1-54331 “ 

15-43315 “ 

154-33159 “ 

1543-3159 “ 


1 Millier = 1000 Kilogrammes = 1 ton sea weight. 


= 2*204737 lbs. avoirdupois. 
0*4535685 KilogTamme 
0-3732223 “ 


Note. —In the new French system, the values o f the base of each measure, viz., 
Metre, Litre, Store, Are, and Gramme, are decreased or increased by the following 
words prefixed to them. Thus, 


Milli expresses the 1000th part. 

Centi “ 100th “ 

Deci “ 10th “ 

Deca (( 10 times the value. 


Hecato expresses 
Chilio 

Myrio M 


100 times the value. 
1000 “ 

1000 “ 


FOREIGN WEIGHTS COMPARED WITH AMERICAN. 


11 ■— - 1 — 


Number 
equal to 


Weights. 

Number 

ei/ual to 

Places. 

Weights. 

100 avoir¬ 
dupois 
pounds. 

Places. 

100 avoir¬ 
dupois 
pounds. 

Aleppo 

Rottoli 

20*46 

Hanover 

Pound 

93-20 

U 

Oke 

3580 

Japan 

Catty 

76-92 

Alexandria. 

Rottoli 

107- 

Leghorn 

Pound 

133-56 

Algiers . 


84- 

Leipsic . 

“ (common) 

97-14 

Amsterdam 
j Antwerp 
Barcelona • 

Pound 

U 

91-8 

96-75 

Lyons 

Madeira . - 

“ (silk) 

98-81 

143-20 

a 

112-6 

Mocha • 

Maund 

33-33 

Batavia • 

Catty 

76-78 

Morea 

Pound 

90-79 

Bengal . 

Seer 

53*57 

Naples . . 

Rottoli 

50-91 

Berlin 

Pound 

96.8 

Rome 

Pound 

133-69 

Bologna . 

U 

125.-3 

Rotterdam . 

a 

91-80 

Bremen . . 

U * ' 

90-93 

Russia . • 

a 

110-86 

Brunswick . 

tc 

9714 

Sicily . • 

a 

142-85 

Cairo • • 

Rottoli 

105. 

Smyrna . . 

Oke 

36-51 

Candia • 

<C 

85-9 

Sumatra . . 

Catty 

35-56 

China 

Catty 

75-45 

Sweden . 

Pound 

106-67 

Constantinople 

Oke 

35-55 

U 

U 

120-68 

Copenhagen 

Pound 

90-80 

Tangiers 

“ (miner’s) 

9 4 - 27 

Corsica . 

« 

131-72 

1 Tripoli • • 

Rottoli. 

89-28 

Cyprus . 

Rottoli 

19-07 

j Tunis • . 

! 

90-09 

Damascus . 


25-28 

Venice . 

Pound (heavy) 

94-74 

Florence 

Pound 

133-56 

U 

“ (light) 

150. 

Geneva . 

“ (heavy) 

82-35 

I Vienna . 

a 

si- 

Genoa 

Hamburgh . 

(C ct 

« a 

92- 86 

93- 63 

1\ arsaw . 

u 

112-25 



























Total Rations per week. 


Provisions. 


77 



Xtl 

c 

3 

o. 

05 

'c 


ta! 

§ 

*<» 


W 

v> 


B 

•o 

o 

P 

CD 

P 


•o 

n 

ST 

o 

*-D 


P* 

CD 




H-4 

1—4 


M 



Beef. 


05 

h-» 



H-4 


j— 1 


Pork. 


H-4 







»4- 

Flour. 

o 















jW"' 




►h* 

Raisins or 
dried fruit. 

p 

pi 

QQ 

M- 

/ 

M-i 



w- 




Pickles. 


uu 


KH 



*V*H 



Rice. 


CO 

CO 

l—^ 


t— 1 

1—4 

1—4 

H- 

1—4 

Biscuits 

• 


H-* 

to 

to 

to 

to 

to 

to 

to 

Sugar. 


h-* 

iNw 


IM-> 

44- 


44- 

44- 


'lea. 

Either; 

o 

-T 

H* 

I-* 

I-* 

h-4 

h-4 

t—* 

M’ 

Coffee. 

P 

B 

o 

<D . 

CO 


f—» 


M 

h-4 

t— 1 

H-* 

H—4 

Cocoa. 





bs3 



to 



Butter. 




to 



to 



Cheese. 


»—* 
Kfr- 




K*- 


kvH 


Beans. 

*1 



►5H 






Molasses. 










Tinegar. 

,5-| 

h-* 

>Nw 


4-fr- 


itM 

itt- 

44- 

•M— 

Spirits. 

Ml 























































78 


Geometry. 


GEOMETEY. 

DEFINITIONS. 

Demonstration is a course of reasoning by which a truth is established. Tt 
consists of, 

Thesis, the truth to be established, and, 

Hypothesis, the foundation for the demonstration. 

Axiom is that which is self-evident and requires no demonstration. 

Theorem is something to be proved by demonstration. 

Postulate is something to be done, but is self evident and requires no demon¬ 
stration. 

Problem is something proposed to be done, and requires demonstration. 
Proposition is either a Theorem or a Problem. 

Corolary is an obvious conseqence deduced from something that has gone 
before. 

Scolium is a remark on preceding propositions, commonly demonstrated by 
algebraical formulae. 

Lemma is something premised for a following demonstration. 

Geometrical Quantities* 

Point is a position, but no magnitude. 

A Line is length, without breadth or thickness. 

A Straight Line is the shortest distance between two points. 

Curved line is a length which in every point changes its direction. 

Superficies, Surface, Area, is that which has length and breadth, but no 
thickness. 

Plane surface is a plane which coincides with a straight line in every direc¬ 
tion. 

Curved surface is a plane which coincides with a curved line. ' 

Solid has length, breadth and thickness. 

Circle* 

Circle, Cirumference, Periphery, is a curved line drawn on a plane surface, and 
bounded at a common distance from one point in the plane, (centre.) 

Radius is a line* drawn from the centre in a circle to the periphery. 

Diameter is a line drawn through the centre to the periphery, or the longest 
line in a circle. - 

Chard is any line extending its both ends to the periphery of a circle, and does 
not go through the centre. 

Arc is a part of a periphery. 

Circle plane, is a plane surface bounded within a circumference. 

Sector is a part of a circle-plane bounded within an arc and two radii. 

Segment is a part of a circle plane bounded within a chord and an arc. 

Zone is a part of a circle included between two parallel chords. 

Lune is the space between the intersecting arcs of two eccentric circles. 

Oval is a round figure having one long and one short diameter at right angles 
to one another. 

Semicircle is a half circle. 

Quadrant is a quarter of a circle. 

Angles* 

Angle is the opening or inclination of two lines which meet in one point. 

If two radii being drawn from the extremities of a circle arc, to the centre; 
the arc, is a measure of the angle at the centre. 

Right angle is when the opening is a quarter of a circle. 

Acute angle is less than a right angle. 

Obtuse angle is greater than a right angle. 

* Line by itself means a straight lino. 

. 

















Geometry. 


T9 


Triangles* 

Triangle is'a figure of three sides. 

Equilateral Triangle has all its sides equal. 
Isosceles Triangle has two of its sides equal. 
Scalene Triangle has all its sides unequal. 
Eight-angled triangle has one right angle. 
Obtuse-angled, triangle, has one obtuse angle. 
Acute-angled triangle has all its angles acute. 


Quadrangles* 

Quadrangle is a figure of four sides. 

Parallelogram haring its opposite sides parallel, and the opposite angles 
equal. 

Square, having its four sides equal, and four right angles. 

Rectangle , having its opposite sides equal, and four right angles. 

Rhombus, having four equal sides, and opposite angles equal hut not right. 

Rhomboid, same as a parallelogram. 

Trapezium, having four unequal sides. 

Trapezoid, having only two opposite sides parallel. 

Gnomon is the space included between the lines forming two similar parallel¬ 
ograms, of which the smaller is inscribed in the larger, so as to have one com¬ 
mon angle. 

Polygons. 

Polygons are plane and rightlined figures. 

Regular Polygons are plane figures which inscribe, or circumscribe a circle, 
and their sides being equal. Polygons are named according to their number of 
sides, thus, 


Trigon has three sides. 

Octagon 

has 

eight 

sides 

Tetragon “ four “ 

Nonagon 

a 

nine 

U 

Pentagon “ five “ 

Decagon 

a 

ten 

a 

Hexagon “ six *• 

Undecagon 

(6 

eleven 

u 

Heptagon “ seven “ 

Dodecagon 

« 

twelve 

u 

For properties of Polygons see page 103. 





Solids* 


Sphere is a solid bounded within a half circle rotating round its diameter. 

Spherical segment, (zone) is a part of a sphere cut off by a plane. 

Spheroid is a sphere flatted or longed at two opposite sides ; as our earth is 
flatted at the poles, and having one diameter shortest; an egg is longed, and 
having one diameter longest. 

Spindle is a solid bounded within a curved line rotating round its base. 

Cylinder is a solid bounded within a rectangle rotating round one of its sides, 
(axis.) A cylinder has a circle plane to its base. 

Cone is bounded within a right-angled triangle rotating round one of its sides 
that forms the right angle. , 

Ungula is the bottom part of a Cone or Cylinder, cut off by a plane passing ob¬ 
liquely through the base. 

Cube is bounded within six squares. 

PuraUdopiped is bounded within six parallelograms. 

Prism is a solid described by a rightlined plane moving in a straight line, so 
that the plane forms an angle to its direction line. 

Prismoid is a prism cut obliquely at the ends. 

Pyramid is bounded between a rightlined plane, and one point at a distance 
from the plane. The sides of the rightlined plane, are bases of triangles deter¬ 
minating at the aforesaid point, (vertex.) 

Perimeter is the sum of all the sides in a figure, plane or solid. 

Polyhedrons . See page 95, for properties and names of the five regular poly¬ 
hedrons 



















8& 


CONSTRUCTIONS. 


A 

i . ^ 

ii. 

To divide a given line AB into 
two equal parts; and to erect a 
perpendicular through the middle. 

s 


J 

/ 

A ! 

D 

\ * 

\B 

2. 

At a given point C on the line 
AB erect a perpendicular CD. 

V C j 

c 

A \ 

1 

/ £ 

3. 

From a given point C, at a dis¬ 
tance from the line AB, draw a per¬ 
pendicular to the line. 



\ 

7 " 

\ 


A'' 


X 


X,_ 

c/ 




i 

/ 

A ! 

/ B 


4. 


At the end A of a given line AB , 
erect a perpendicular AC. 


5. 


Through a given point C, at a 
distance from the line AB, draw a 
line CD parallel to AB. 



6 . 


On the given line AB, and at the 
point B, construct an angle, equal 
to the angle CDE. 






























Constructions. 


81 











































83 


Constructions. 


A 

! M 

. v \ 

13. 

Through a given point A in a 
circumference, draw a tangent to the 
circle. 

\ \ 

V 

/ 

/ / 
y 

» N y 

S'7 

A 

\ 

cix 

14 . 

From a given point A out of a 
circumference, draw a tangent to the 
circle. 

M- . Xc jj'\ 

15 . 

Draw a circle with the given radius 

C D' , that will tangent the circle AB 
at C 

.S C —0 

4 ^ 

jv^y vn/^ n 

' jMr\. N 

16. 

Draw a circle with the given radius 
CD, that will tangent the two circles 

A and B. 



x. 

% X 

* v\ \ 

T'Jx. > 

/Xlo; 

'—^B 

17. 

Draw a tangent to the two circles 

A and B, that is on one side of them. 


c i \/ \ 

\M V A 

18. 

Draw a tangent between two 
circles A and B. • 

*----- 

r * 

V S 

B ” 

M } 1 

\ fySO* / 

x ^ — 

J '% 


























CoNSTRTSOriONS. 


83 


T 

X 'mfe' 

A 'A^L B 

19. 

With a given radius r, draw a circle 
that will tangent the given line AB 
and the given circle CD. 

0 X2>CW T B 

20. 

To find the centre and radius of a 
circle that will tangent the given 
circle AB at C, and the line DE. 


» 

[p \r 

V s'* f 

A/%1 

21. 

To find the centre and radius of 
a circle, that will tangent the given 
line AB at C, and the circle DE. 

(A ■ 

\/ Jy^r 
•: ' - ' -V:: B 

22. 

To find the centre and radius of a 
circle that will tangent the given line 
AB at C, and the circle DE. 


23. 

Between two lines, draw two circles j 
that will tangent themselves and the 
lines. 

I 

i 

L— 1 

24. 

Draw a circle-arc, that will tangent 
two lines inclined to one another, and 
the one tangenting point A being 
given. 






























Constructions. 


*4 


\ 

V,>*' ?: 

ft* / 

iVv? 

2 

25. 

Draw a circle-arc that will tangent 
two lines, and go through a given 
point C, on the line which, besects 
the angle of the lines. 

** A 

B 

26. 

To draw a Cyma, or two circle 
arcs that will tangent themselves, and 
two parallel lines at given points A 
and B. 

'-f 

—•"t-k 

J 

27. 

To draw a Talon , or two circle 
arcs that will tangent themselves ; and 
meet two parallel lines at right angles, 
in the given points A and B. 

tfzx 

*ty0:y 

^...2 

£ 

K f 

A 

/>f 

28. 

To draw a circle-arc without re¬ 
course to its centre but its chords and 
height being given. 



29. 

To find the centre and radius of a 
circle that will tangent the three 
sides in a triangle. 


30. 

To inscribe an equilateral triangle 
in a given circle. 




























Constructions. 


85 































86 


Constructions. 











































Constructions. 


87 

































88 


Circle. 


CIRCLE. 


The periphery of a circle is divided into 360° (degrees) equal parts, each called 

a degree. 

One degree = 60' (minutes.) 

One minute = 60" (seconds.) 

Half a circle (hemisphere) = 180°. 

Quarter of a circle (quadrant) = 90°. 

By the accompanying formula any part of the circle can he calculated. 

f ormula tor tne circle. 


P = 7rd = 3-14(2,.1, 

r = a / - = 0 - 564p'a, - - 

- 7, 

p = 2 7rr = 6-28r,.2, 

V 

a=~=-- 0-785<2 2 , - - - . 

4 

• 8, 

p = 2 y-^a = 3 - 54]/a, - - - - 3, 

a = ?rr 2 = 3*14r 2 , - - . . 

* 9, 

d=* = -*L, .4, 

p 2 p 2 

4 a- 12-56’ 

10, 

d = 2 -^/ r ~ = vl2S - * 5 > 

1 

• 

• 

t 

• 

ns f 

II 

II 

13 

- 11, 

„ P P a 

2 71 6-28’. 6 ’ 

a = - 1.57 rd, - - - - 

2 5 

12. 


7t = 3-141592653589793238462643383279502884197169399375105820974944 5923078 
164062862089986280348253421170679821480865132823066470938446. 


%t = 6-28218530710000. 

3^ = 9-42477796070000. 


- = 1-27323954473480. 

4rr = 12-5663706143000. 

5^- = 15-7079632679000. 

6w = 18-8495559215000. 

7tt = 21-9911485751000. 

8*- = 25-1327412887000. 

9- = 28-2743338823000. 

= 1-57079632679000. 

±7T = 0-78539816339700. 
a*- = 1-04719755119600. 

y Q - = 0-52359877559800. 


3 

- = 0-95492965855110. 

7T 

- = 1-90965931710220. 

7T 

12 

— = 3-81971863420440. 

71 

360 

— = 114-591559026122. 

7T 

i^r = 0-39269908169800. 


tt 2 = 9-86965000000000. 

y v fr == 0-26179938779900. 

= 0-00872667621060. 

<J60 

1 = 0-31830988618370. 

7T 

i 

= 1-77245300000000. 
y 7T 

\/- = 0-56418900000000. 

7T 

- = 0-63661977236740. 

5T 


— =0-07957747154500. 

4w 


Letters denote, 

r = radius of the circle. 
d = diametei-. 
p = periphery. 

a = area of a circle, or part thereof. 

5 = circle-arc, length of. 

c = chord of a segment, length of. 
h — height of a segment, 
s = side of a regular polygon. 
v = centi-e angle. 
w = polygon angle. 

j0bg=-Be careful to express all the dimensions by the same uuit,as miles, rods, 
yards, feet, or inches, &c., &c., or else the calculation will be wrong. 

Example 1. Fig. 49. The diameter of a circle is 8 feet, 8 inches, how long is the 
fib cumference ? 

Formula 1. p = nd — 3-14X8-666 = 27-211 feet, the answer. 


J 















Longemetry, 


80 



8 * 





































90 Loxgemetry. 















































Loxgemetry. 


91 



























































92 


Longemetry. 




c 3 = a? + b\ 
a* = c 3 - 6 3 , 
b* = d* - a\ 


c 3 = +£ 2 — 2&d, 

A = %/ a 3 — / 7 3 . 

j _ a 3 + £ 3 — c 3 . 


c 2 = a 2 + 2> 3 + 2£d, 


A 2 = V a 3 — d 3 , 




A : c, 
ad 
c 



a : c = d : [b — d), 

, ab 

(l —-? 

c + a 

r = i?. 



72 . 


a : c = b : d, 
ad = be. 





























Longemetry. 


93 



1 

1 > 

k A 

__ 1 

\ 1 

1 

\ / 

> 

/ • 

\ V 

/! 

\/ S 

!\ 

l | 

( _- r _ 


v 

-- y~ - 

--7J / 

v. 1 

1 ' 

\ * 

\i 

/>' \ 

/ % 

1 ' 
r/ 

• S 

i ^ 

£ 

i 

i 

















































94 


Longemetry. 

































POLYHEj/ftONS. 


PS 



Tetrahedron. 

r = 0-20413 s. 

R = 0-60725 §. 
a «* 1-73205 s 2 . 
c = 0-11785 s>. 


Hexahedron. 

7- = 0*50000 s. 

£ = 0-86602 g. 

a = 6-00000 $3. 
c = 1-00000 s*. 


Octahedron. 

r = 0-40721 s. 
R = 0-71710 s. 
a = 3-46410 s 2 - 
c = 0-47140 s* 



Dodecahedron. 

r = 1-11350 s. 
R = 1-36428 s. 
tEL = 20-5457 sa. 
c = 7-66312 S’. 


Icosahedron. 

r = 0*7558 g. 
R = 0-9510 s. 
a = 8-66025 s a . 
c = 2-18169 s 3 . 


r = Radius of an inscribed Sphere. 

R = Radius of circumscribed Sphere, 
a = Area of the Polyhedrons. 
c = Cubic contents of the Polyhedrons. 
S = Side or edge of the Polyhedrons. 


J 





































96 


Planemetry. 




Triangle. 

bh ., . 

£ 3 . — ^ — ^b h, 


90. Square. 

a = s 9 = U\ 
a = O'lQlliT = 2-8284 c\ 


91. Rectangle. 

a = a bf 
a = b V d 1 - 1 


a = 


94. Quadrangle. 

a = £/*(« + 6). 

95. Quadrangle. 

+ b h’ + c h). 
































PLA.NEMETRY. 


97 



96. Circle Plane. 
a = n r* U 0-785 d\ 
a = ^= 0.0794 P\ 

97. Circle Ring. 

a 17r(i ? 2 — r 3 ) = n(R + r )(R~r), 
a = 0-785(2> 3 — P). 

98. Sector. 



a = bbr, 

Tt r 3 v v 

a "860 -3 TliT* 


Segment. 


a = s{3 r — c (r — /i)], 


n r" 1 v c , r. 
a = "360" + 2^ r 


Quadrant. 

a = 0-785 r 3 = 0-3916 c 3 


a = 0-215 r 3 = 0-1075 e 3 . 


9 





















98 


Planemetry. 



102 . 


Fllipse. 


a = 7t R r = 0-785 D d. 



103. 


Parabola. 


a = f b h =*= £ 6 s , 


a = i hVph. 



104. Irregular Figure. 


&= b(h + h' + h"). 


105. 



Ellipsoid. 


a = 8-88 r V R 1 — r a , 


a = 2-22 d VT) 1 - d\ 


106 . 



Cylinder. 

a == 2 n r h — n d h, 


a 


2 nr n d 



The road to Extremity of Space. 

1st. Draw a Circle and inscribe a Square, and in 

that Square a Circle, &c., &c., &c. The last 

figure that can he drawn, is one extremity of Space 
Required if the last one is a Circle or a Square? 

2nd. Draw a Circle and circumscribe a Square, 

and around that Square a Circle, Ac., &c.. &c . 

The last one that can be circumscribed is the other 
extremity of space. Required if the last figure is 
a Circle or a Square? 























































99 


Surface or ?~lids. 



109, Sphere Sector. 



110. Circle Zone. 



'ill. Cone. 






































100 


Stereometry. 



113. 


c = 


Sphere. 

- 4-189 r% 


= 0-523 d\ 


114. For us, 


C = 2 rti R r 2 = 19-72 R r a , 
c - 2-463 D d\ 


115. Sphere Sector. 

€ = I rt r 2 h = 2-0944 r 2 h, 


c = | ti r 2 (r + v/r 2 - i c 2 ). 


116. Zone. 

C n A 2 (r — | A), 

c 3 + 4 A 2 


Conic Frustrum. 
g rt A( J R 2 -t 1? r + r 2 ), 
A(Z) a + Dd + d') 


118. 


c = 


c = 













































121, Paraboloid. 

c = b n i" x h = 1-5707 r* A. 


Wedge Frustrum. 

h s. .. 

C = -g (a + b). 


122. Pyramid. 

c = ia h, 


ns h / „ s a 

= 6 \/ r ~ r’ 


Pyramidic Frustrum. 


a + V A a). 


Stereometry. 


119. Cylinder. 

C = hr' h - 0.785 d“ h. 


c-^-0 0796 ®* 5. 


120. Ellipsoid. 

c = 0-424 rf R r* = 4-1847 £ r% 
c = 0 053 n' D d* = 0-5231 D d*. 


9 * 



































IVZ 


STEREOMETRY. 



125. Cask, 

c _ 1-0453 i(0-4 D“+ 0-2 D d + 0-15<e), 

Gallon = ~{4 D* + 2 + 1-5 <f )• 



' «S«W®»>65fflS5 


126. Cylinder Sections. 

C = 7tr*(l + V — f ?•), 

c = it r\l + I') — 2*1 r 3 



127. 


Circular Spindle. 


C = 7t(e c 8 — 0-2 d[c+§ V c 2 + o f2 ] V d? + c s ) 


Example 1. Fig. 92. The base of a Triangle is b = 8 feet, 3 inches, and the 
height, h — 5 feet, 6 inches. What is the area a = ? 


b h 8-25 X 5-5 


= 22'6875 square feet. 


2 2 

Example 2. Fig. 98. A Circle Sector having an angles = 39° and the radius 
r = 67^ inches. What is the area of the sector a = ? 

TT.r^v 3-J4X 67-75*X 39° 


a = 


360 


360 


= 1562T square feet. 


Example 3. Fig. 110. A Spherical Zone having its diameter c = 18j inches 
and height h = 7f inches. What is the convex surface of the Zone ? 


a =~(c* -j-h 2 ^ ~~J~ ^18-5* -f 7 - 75 2 ^ = 315-96 square inches. 

Example 4. Fig. 8«. Require the radius A* of a Sphere that will circumscribe 
a Dodecahedron with the side s = 9 inches. 

R = 1-36428 X 9 = 12-27852 inches, the answer. 

Example 5. Fig. 118. A Frustrum of a Cone having its bottom diameter D = 13 
inches, the top diameter d — 5£ inches, and the height h = 25 inches. What is 
the cubic contents c = ? 

c = Aj Tl. h(D* + Dd + 0-2618 X 25 (l3* + 13 X 5-25 + 525*)= 20995 

cubic inches. 

Example 6. Fig. 125. A Cask having its bung diameter D — 36 inches, head 
diameter d = 28 inches, and length l — 56 inches, (inside measurement) how 
many gallons of liquid can be contained in the cask ? (The gallon = 231 cub. in.) 

Gallon = 2 -2 oq(4 X 36* + 2 X 36 X 28 + 1-5 X 28*)= 214 gallons. 




































wEOMETRY.-TABLE OF POLYGONS. 


W 


Example 7. Fig. 50. Require the length of the circle-arc b, when the angle 
v — 42P, and the radius r = 4 feet, 3 inches ? 

_rrv_ 3-14X4-25X42 
b lg0 18Q 3113 feet. 

Example 8. Fig. 52. Require the radius of a circle-arc, whose chord is 9 feet, 
4 inches, and height, h — 1 foot, 8 inches ? 


r = 


c 2 + 4 ft 2 9-33 2 -j-4Xl'66 3 98-0711 


= 7-384 feet. 


8h 8XF66 13-28 

Example 9. Fig. 68. The three sides in a triangle being, a = 6*42, b = 7-75, 
and c = 8-66 feet. How high is the triangle over the base b ? 


d = 


a 2 -t-6 2 —c 2 6-42 2 -f7-75 2 — 8-66 2 


26 


2X8-66 


= 1-5175 feet, 


the height h — -j/a 2 — d% = }/6.42 2 —1-5175 2 = 6-24 feet, the answer. 

Example 10. Fig. 77. The radius of a walking beam is, r — 8‘36feet, the stroke 
S — 5"5 feet. How much is the vibration V= ? 


Vibration, 


V=r— — £1 = 8-36 — 8-36 2 — 


n 2 


=0-471 feet 


5-65 inches = 5-—, the answer. 

32 


TABLE OF POLYGONS. 


Number 
o: sides 

in the 
Polygon. 

Trigon. 

3 

Tetragon. 

4 

Pentagon. 

5 

Hexagon. 

6 

Heptagon. 

7 

Octagon. 

8 

Nonagon. 

9 

Decagon. 

10 

Undecagon. 

11 

Dodecagon. 

12 


14 


15 


16 


18 


20 


24 


Polygon 
Angle v. 



Side 
= k R. 


-- h S*. 



1-732 

1-4142 

1-1755 

1-0000 

0-8677 

0-7653 

0-6810 

0-6180 

0-5634 

0-5176 

0-4450 

0-4158 

0-3900 

0-3472 

0-3130 

0-2610 



0-4330 

1-0000 

1- 7205 

2- 5980 

3- 6339 

4- 8284 
6-1820 
7-6942 
9-3656 

11-196 

15-334 

17-642 

20-128 

25-534 

40-634 

45-593 


Apo’em Side 
= k R. = ft r. 



0-5000 

0-7071 

0-8090 

0-8660 

0-9009 

0-9238 

0-9396 

0-9510 

0-9595 

0-9659 

0-9762 

0-9781 

0-9807 

0-9848 

0-9877 

0-9914 



3-4641 

2-0000 

1-4536 

1-1547 

0-9631 

0-8284 

0-7279 

0-6498 

0-5872 

0-5359 

0-4562 

0-t250 

0-4068 

0-3526 

0-3166 

0-2632 


Area 
= A r 2 . 



5-1961 

4-0000 

3-6327 

3-4640 

3-3710 

3-3130 

3-2750 

3-2490 

3-2290 

3-2152 

3-1935 

3-1882 

3-1824 

3-1737 

3-1676 

3-1596 


Explanation of tlic Table for Polygons. 

The number of sides in the polygon is noted in the first column. 
lc = tabular coefficient, to be multiplied as noted on the top of the columns. 
Example 1. How long is the side of an inscribed Pentagon, when the radius 
of the circle is 3 feet, and 4 inches ? (4 inches = 0-333 feet.) 

3-333Xl'1755 = 3-9179 feet, the answer. 

Example 2. What is the area of a Heptagon when one of its sides is 13-75 inches 
13 - 75 a X3-6339=687-02 square inches. 


J 























































104 


.Circumferences and Areas of Circus. 



Circ. 

/—X 

Area. 


Circ. 

Area 

Diame- 

Circ. 

Diame¬ 

ter. 

o 

IIP 

Diame¬ 

ter. 

o 

in 

ter. 

0 

32* T 

-) *0981 

•00076 

5— 

15*70 

19*635 

.11 — 

34.55 

t 7 b 

*1963 

.00306 


-16*10 

20*629 


- 34*95 

i -- 

*3926 

*01227 

i 7 

-16*49 

21*647 

i - 

- 35*34 

3 

1 (T 

*5890 

*02761 


16*88 

22*690 


J 35*73 

i — 

*7854 

*04908 

i- 

17*27 

23*758 


36*12 

ft 

*9817 

*07669 


17*67 

24*850 


- 36*52 

a - - 

1*178 

*1104 

1 - 

18*06 

25*967 

s - 

- 36*91 

7 

TTi 

1*374 

*1503 


18*45 

27*108 


37*30 

i — 

1*570 

*1963 

6 — 

18*84 

28*274 

12— 

37*69 


1*767 

*2485 


19*24 

29*464 


38*09 

a - 

1*963 

*3067 

i - 

19*63 

30*679 

i - 

38*48 

u 

2*159 

*3712 


20*02 

31*919 


38*87 

z — 

2*356 

*4417 

4— 

20*42 

33*183 

i 

J 39*27 

H 

2*552 

*5184 


20*81 

34*471 


h 39*66 

i - 

2.748 

*6013 

Z - 

21*20 

35*784 

1 

40*05 

is 

2*945 

*6902 


21*57 

37*122 


40*44 

l 

3*141 

*7854 

7 — 

21*99 

38*484 

13- 

J 40*84 


3*534 

*9940 


22*38 

39*871 


41*23 

i - 

3*927 

1*227 

4 - 

22*77 

41*282 

i ~ 

J 41*62 


J 4*319 

1*484 


^23*16 

42*718 


J 42*01 

i — 

44*712 

1*767 

4— 

23*56 

44*178 

i 

42*41 


5*105 

2*073 


323*95 

45*663 


1 42*80 

Z - 

5*497 

2*405 

Z ~ 

24*34 

47*173 

z - 

43*19 


-5*890 

2*761 


24*74 

48*707 


43*58 

a- 

-6*283 

3*141 

8 — 

25*13 

50*265 

14-[ 

J 43*98 


6-675 

3*546 


25*52 

51*848 


44*37 

i - 

7*068 

3*976 


25*91 

53*456 

i - 

44*76 


47*461 

4*430 


26*31 

55*088 


-45*16 


-7*854 

4*908 

*--- 

26*70 

56*745 

h 

J 45*55 


8*246 

5*411 


327*09 

58*426 


45*94 

Z - 

->8*639 

5*939 

Z -- 

27*48 

60*132 

z - 

-46*33 


,9*032 

6*491 


27-S8 

61*862 


46*73 

3__ 

9*424 

7*068 

9— 

28*27 

63*617 

15- 

47*12 


9*817 

7*669 


28*66 

65*396 


47*51 

i J 

1 10*21 

8*295 

4 J 

29*05 

67*200 

i - 

47*90 

i— 

110*60 

8*946 


29*45 

69*029 


48*30 

,10*99 

9*621 

4— 

29*84 

70*882 

4- 

48*69 


11*38 

10*320 


30*23 

72*759 


49*08 

Z - 

11*78 

11*044 

1 

30*63 

74*662 

1 - 

49*48 


12*17 

11*793 


31*02 

76*588 


49*87 

4 — 

-12*56 

12*566 

10—. 

31*41 

78*539 

16— 

50*26 


12*95 

13*364 


31*80 

]32*20 

80*515 

82*516 


50*65 

51*05 

i - 

-13*35 

14*186 

4 - 

4 - 


13*74 

15*033 


32*59 

84*540 


51*44 

A_i 

14*13 

15*904 

4- - 

32*98 

86*590 

4- 

51*83 


14*52 

16*800 

• . 

33*37 

88*664 


52*22 

Z 

-14*92 

17*720 

Z - 

33*77 

90*762 

z - 

52*62 


-15.31 

18*665 


34*16 

92*885 


53*01 







- J. 



Area. 



95*033 
97*205 
99*402 
101*62 
103*86 
106*13 
108*43 
] 10*75 
113*09 
115*46 
117*85 
120*27 
122*71 
125*18 
127*67 
130*19 
132*73 
135*29 
137*88 
140*50 
143*13 
145*80 
148*48 
151*20 
153*93 
156*69 
159*48 
162*29 
165*13 
167*98 
170*87 
173*78 
176*71 
179*67 
182*65 
185*66 
188-69 
191*74 
194*82 
197*93 
201*06 
204*21 
207*39 
210*59 
213*82 
217*07 
220*35 
223*65 



































Circumferences and Areas of Circles. 105 



Circ. 

Area. 


Circ. 

Area 


Circ. 

Area. 

Diame- 

r ^ 

Jill 

Diame- 


/flU; 

Diame- 



ter. 

KJ 


ter. 

W 

l|j|' 

ter. 

o 

iHpi 

17 —r 

53*40 

226-98 

23-i 

72-25 

415*47 

29-r 

91-10 

660-52 


53-79 

230-33 


72-64 

420-00 


91-49 

666-22 

i - 

54-19 

233-70 

£ " 

73-04 

424*55 

£ -• 

91-89 

671-95 


54-58 

237-10 


73-43 

429*13 


92-28 

677-71 

i— 

54-97 

240-52 

i— 

73-82 

433-73 

*4 

92-67 

683-49 


55-37 

243-97 


74-21 

438*30 


93-06 

689-29 

2 

4 : 

55-76 

247-45 

| -- 

74-61 

443-01 

s - 

93-46 

695-12 


56-16 

250-94 


75- 

447*69 


93-85 

700-98 

18— 

56-54 

254-46 

24— 

75-39 

452-39 

30— 

94-24 

706-86 

j- 

56-94 

258-01 


75-79 

457-11 


94-64 

712-76 

i 4 

57-33 

261-58 

£ - 

76-18 

461-86 

£ - 

95-03 

718-69 

i 

1 57*72 

265-18 


76*57 

466-63 


95-42 

724-64 

i 

-158-11 

268-80 

i— 

76-96 

471-43 

£ L 

195*81 

730-61 


- 58-51 

272-44 


77-36 

476-25 


96-21 

736-61 

i - 

-58-90 

276-11 

1 - 

77-75 

481-10 

3 

96-60 

742-64 


-159-29 

279-81 


78-14 

485-97 


96-99 

748-69 

19- 

-59-69 

283-52 

25— 

78-54 

490-87 

31 " 

; 97-38 

754-76 


-60-08 

287-27 


78-93 

495-79 


97*78 

760-86 

£ - 

60-47 

291-03 

£ 

79-32 

500-74 

£ j 

'98-17 

766-99 


- 60-86 

294-83 


79-71 

505-71 


1 98-56 

773-14 

4- 

- 61-26 

298*64 

i— 

■>80*10 

510-70 

V 

■98-96 

779*31 


- 61-65 

302-48 


■80*50 

515-72 


99-35 

785-51 

1 - 

- 62-04 

306-35 

3 - 

180-89 

520-70 

s - 

99-74 

791-73 


62-43 

310-24 


-81-28 

525-83 


100-1 

797-97 

20- 

- 62-83 

314-16 

26-4 

-81-68 

530-93 

■32—- 

-100-5 

804*24 


- 63-22 

318-09 


-82-07 

530-04 


100-9 

810*54 

4 - 

- 63-61 

322-06 

£ - 

82-46 

541-18 

£ ~ 

101*3 

816-86 


- 64-01 

326-05 


-82-85 

546-35 


-101-7 

823-21 

4- 

- 64.40 

330-06 

i- 

-83-25 

551-54 


102-1 

829-57 


-i 64*79 

334-10 


-83-64 

556-76 


102-4 

835-97 

1 - 

-'65-18 

338-16 

1 - 

-84-03 

562-00 

•3 - 

102-8 

842-39 


-65-58 

342-25 


84-43 

567-26 


103-2 

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r 65-97 

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350-49 


- 85*21 

577-87 


104- 

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4 -i 

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£ - 

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67-15 

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r 67-54 

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67-93 

367-28 


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3 - 

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371-54 

3 - 

-87-17 

604-80 

1 *■ 

-106* 

894-61 


-68-72 

375-82 


-87-57 

610-26 


106-4 

90-1*25 

22- 

-'69-11 

380-13 

28- 

-87-96 

615-75 

34- 

-106-8 

907-92 


-69-50 

384-46 


-88-35 

621-26 


107*2 

914-61 

4 - 

469-90 

388-82 

£ - 

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JL - 

4 

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921-32 


J 70-29 

393-20 


-J 89* 14 

632-35 


■j 107*9 

928-06 

4- 

J 70-68 

397-60 


-89-53 

637-94 

£~ 

- 108-3 

934-82 


-71-07 

402-03 


-89-92 

643-54 


- 108-7 

941-60 

4 

-71-47 

406-49 

3 - 

-90-32 

649-18 

5 - 

' 109-1 

948-41 


-71-86 

410-97 


-90-71 

654-83 


- 109-5 

955-25 




— 






































1C6 


Circumferences and Areas of Gin c ues. 



Circ. 

Area. 


Circ. 

Diame¬ 

ter. 

o 

up 

Diame¬ 

ter. 

c 

35 -r 

109*9 

962-11 

41 —j 

128.8 


110*3 

968-99 


129-1 

i 

110-7 

975-90 

i - 

129-5 


111*1 

982-84 


129-9 


111*5 

989-80 

4— 

130-3 


111*9 

996-78 


130-7 

2 

112-3 

1003-7 

1 - 

131-1 


112-7 

1010-8 


131-5 

36— 

ii a* 

1017-8 

42— 

131-9 


113-4 

1024-9 


132-3 

i - 

113-8 

1032-0 

i - 

132-7 


1114-2 

1039-1 


133-1 

h - 

114-6 

1046-3 


133-5 


115- 

1053-5 


133-9 

1 ' 

115-4 

1060-7 

s - 

134-3 


115-8 

1067-9 


134-6 

37- 

116-2 

1075-2 

43 -r 

135- 

-1116*6 

1082-4 


135-4 

i " 

-1,117* 

1089-7 

i - 

135-8 


J 117*4 

1097-1 


136-2 

i—j 

J117-8 

1104-4 

4— 

" 136-6 


118-2 

1111-8 


137- 

2 - 

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1119-2 

1 - 

-137*4 


118-9 

1126-6 


137-8 

38- 

-119-3 

1134-1 

44—r 

138-2 


4119-7 

1141-5 


-138-6 

i - 

120-1 

1149-0 

i - 

139- 


-120-5 

1156-6 


139-4 

i- 

-120-9 

1164-1 

4— 

139*8 


-121*3 

1171-7 


140-1 

2 - 

J 121-7 

1179-3 

i 4 

140-5 


-122-1 

1186-9 


140-9 

39 - 

-122-5 

1194-5 

45 — 

141-3 


- 122-9 

1202-2 


141-7 

i - 

123-3 

1209-9 

l - 

-142-1 


-,123-7 

1217-6 


142-5 

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h 124- 

1225-4 

4- 

142-9 

124-4 

1233-1 


143-3 

2 - 

-124-8 

1240-9 

1 - 

143-7 


- 125-2 

1248-7 


-144-1 

40- 

- 125-6 

1256-6 

46- 

- 144-5 


-126- 

1264-5 


J 144-9 

2 - 

n 126-4 

1272-3 

i - 

-'145*2 


-126-8 

1280-3 


-1 145*6 

i-, 

-127-2 

1288-2 

4- 

-146- 


- 127-6 

1296-2 


- 146-4 

2 

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-128-4 

1312-2 


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— 

Area 


Circ. 

Area. 


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ter. 

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1320-2 

47- 

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1734-9 

1328-3 


-148- 

1744-1 

1336-4 

i - 

- 148-4 

1753*4 

1344-5 

J 148-8 

1762-7 

1352-6 

i-' 

149-2 

1772-0 

1360-8 


149-6 

1781-3 

1369-0 

s - 

150- 

1790-7 

1377-2 


150-4 

1800-1 

1385-4 

48-— 

150-7 

1809-5 

1393-7 


150-1 

1818-9 

1401-9 

i - 

151-5 

1828-4 

1410-2 


151-9 

1837-9 

1418-6 

i , 

1152-3 

1847-4 

1426-9 


152-7 

1856-9 

1435-3 

1 

153-1 

1866-5 

1443-7 


153-5 

1876-1 

1452-2 

49 - 

153-9 

1885-7 

1460-6 


154-3 

1895-3 

1469-1 

i - 

154-7 

1905-0 

1477-6 


{155-1 

1914-7 

1486-1 

i 

1155-5 
155-9 

1924-4 

1494-7 


1934-1 

1503-3 

2 

156-2 

1943-9 

1511-9 


156-6 

1953-6 

1520-5 

50 —- 

157- 

1963-5 

1529-1 


157-4 

1973-3 

1537-8 

i - 

157*8 

1983-1 

1546-5 


158-2 

1993-0 

1555-2 

4- 

J 158-6 

•2002-9 

1564-0 


159- 

2012-8 

15-72-8 

2 - 

159-4 

2022-8 

1581-6 


159-8 

2032-8 

1590-4 

51- 

160-2 

2042-8 

1599-2 


160-6 

2052-8 

1608-1 

i - 

161- 

2062-9 

1617-0 


161-3 

2072-9 

1625-9 

4 - 

161-7 

2083-0 

1634-9 

162-1 

2093-2 

1643-8 

2 - 

162-5 

2103-3 

1652-S 


162-9 

2113-5 

1661-9 

52 

163-3 

2123-7 

1670-9 


163-7 

2133-9 

1680-0 

4 

il64-l 

2144-1 

1689-1 


164-5 

2154-4 

1698-2 

4- 

164-9 

2164-7 

1707-3 


165-3 

2175-0 

1716-5 

2 * 

165-7 

21S5-4 

1725-7 


166-1 

2195-7 







































Circumferences and Areas of Circles. 107 



Circ. 

Area. 


Circ. 

A -»• r* | 


Circ. 

Area. ] 

Diame¬ 

ter. 

c3 


Diame¬ 

ter. 

O 

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Diame¬ 

ter. 

c 

U 1 

53 -r 

166-5 

2206*1 

59 - 

185-3 

2733-9 

65 —r 

204-2 

3318*3 


166-8 

2216-6 


185-7 

2745-5 


204-5 

3331-0 

i 

167-2 

2227-0 

i -- 

186-1 

2757*1 

i - 

204*9 

3343*8 


167-6 

2237-5 


186-5 

2768-8 


205-3 

3356*7 

* — 

16S- 

2248-0 

i— 

186-9 

2780-5 

h 

205*7 

3369-5 


168-4 

2258*5 


187-3 

2792-2 


206*1 

3382-4 

s - 

168-8 

2269-0 

1 - 

1S7-7 

2803-9 

1 - 

206*5 

3395*3 

169-2 

2279*6- 


188*1 

2815-6 


206-9 

3408*2 

54—p 

169-6 

2290-2 

60— 

188*4 

2827-4 

66 —t 

207-3 

3421-2 


L70- 

2300-8 


188*8 

2839-2 


207-7 

3434*1 

i - 

170-4 

2311-4 

i T 

189-2 

2851-0 

i - 

208-1 

3447*1 


170-8 

2322-1 


1896 

2862-8 


208-5 

3460*1 

i— 

171-2 

2332-8 


190 

2874-7 


208-9 

3473-2 


171-6 

2343-5 


190 4 

2886-6 


209-3 

3486-3 

i - 

172- 

2354-2 

1 - 

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209-7 

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172-3 

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191-2 

2910-5 


210* 

3512-5 

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172-7 

2375-8 

61 — 

191-6 

2922-4 

67 - 

210-4 

3525-6 


J 1731 

2386-6 


192* 

2934-4 


210-8 

3538-8 

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J 173-5 

2397*4 

i + 

192-4 

2946-4 

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211-2 

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J 173-9 

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'192-8 

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193-6 

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194-3 

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213-2 

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2463-0 

62—r 

194-7 

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213-6 

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195-1 

3031-2 


214* 

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195*9 

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214-8 

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196-3 

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4 177-8 

2518*2 


196-7 

3080-2 


215-5 

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J 178-2 

2529-4 

1 -- 

197'1 

3092-5 

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215-9 

3712-2 


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2540-5 


197-5 

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216-3 

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- 179-4 

2562-9 


198-3 

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217*1 

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179-8 

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199- 

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217-9 

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4- 

199-4 

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218*3 

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2608-0 


199-8 

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218*7 

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1 - 

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4 - 

-200-2 

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219*1 

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2630-7 


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- 219*5 

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58 — 

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2642-0 

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201*4 

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220-3 

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2676-3 


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221* 

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221-4 

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h 184*1 

2699-3 


203- 

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221*8 

3917-4 


r 184*5 

2710-8 

4 - 

203-4 

3292-8 

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222-2 

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j 184-9 

2722-4 


203-8 

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222-6 

3945-2 










































Circumferences and Areas of Circles. 



Circ. 

Area. 


Circ. 

Area. 


Circ. 

j Area. 

Diame- 

o 


Diame- 

^ ' 

i mu■, 

Diame- 


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ter. 

O 

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ter. 


ijp 

ter. 

w 

lii 

71 -n 

223- 

3959-2 

77 —I 

241-9 

4656-6 

83- 

260-7 

5410-6 

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223-4 

3973-1 


242-2 

4671-7 


i261-l 

5426-9 

i -- 

223-8 

39S7-1 

X -- 

4 

242-6 

4686-9 

X - 
4 

-261*5 

5443-2 


224-2 

4001-1 


243- 

4702-1 


-261-9 

5459-6 

i— 

224-6 

4015-1 

i— 

243-4 

4717-3 


262-3 

5476-0 


225- 

4029-2 


243-8 

4732-5 


262-7 

5492-4 

1 - 

225-4 

4043-2 

i - 

244-2 

4747-7 

1 -i 

263-1 

5508-8 


225-8 

4067-3 


244-6 

4763-0 


263-5 

5525-3 

72— 

226-1 

4071-5 

78— 

245- 

4778-3 

84— 

263*8 

5541-7 


226-5 

4085-6 


245-4 

4793-7 


264-2 

5558-2 

i - 

226-9 

4099-8 

JL 

4 

245-8 

4809-0 

i - 

264-6 

5574-8 


227-3 

4114-0 


246-2 

4824*4 


265- 

5591-3 

4—- 

227-7 

4128-2 

i-f 

246-6 

4839-8 

i [ 

265-4 

5607-9 


228-1 

4142-5 


247- 

4855-2 


265-8 

5624-5 

1 - 

228-5 

4156-7 

1 - 

247-4 

4870-7 

1 

266-2 

5641-1 


228-9 

4171-0 


247-7 

4886-1 


266-6 

5657-8 

73-j 

229-3 

4185-3 

79-- 

248-1 

4901-6 

85- 

267* 

5674-5 


-'229-7 

4199-7 


248-5 

4917*2 


267-4 

5691-2 

4 * 

230-1 

4214-1 

J - 

248-9 

4932-7 

i " 

267-8 

5707-9 


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4228-5 


249-3 

4948-3 


268-2 

5724-6 

i- 

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4242-9 

4— 

249-7 

4963-9 

4 - 

269-6 

5741-4 


-J 231-3 

4257-3 


250-1 

4979-5 


268-9 

5758-2 

1 - 

231-6 

4271-8 

S - 

250-5 

4995-1 

1 - 

269-3 

5775-0 


-|232- 

4286-3 


250-9 

5010-8 


269-7 

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74— 

232-4 

4300-8 

80—r 

251-3 

5026-5 

86 — 

270-1 

5S08-8 


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4315-3 


251-7 

5042-2 


270-5 

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4 - 

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252-1 

5058-0 

X 

4 

270-9 

5842-6 


233-6 

4344-5 


252-5 

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271-3 

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4- 

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4359-1 

4— 

252-8 

5089-5 

4- 

271-7 

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272-1 

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1 - 

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4388-4 

1 - 

253-6 

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1 - 

272-5 

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-235-2 

4403-1 


254- 

5137-1 


272-9 

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- 235-6 

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254-4 

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273-3 

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273-7 

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255-6 

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274-4 

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4- 

A 237-1 

4476-9 

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256- 

5216-8 

i - 

274-8 

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4491-8 


256-4 

5232-8 


275-2 

6030-4 

1 - 

237-9 

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1 - 

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275-6 

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276-4 

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-239-1 

4551-4 


-258- 

5297-1 


- 276-8 

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4 - 

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4 - 

1258-3 

5313-2 

4 - 

- 277-2 

6116-7 


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4581-3 


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5329-4 


- 277-6 

6134-0 

4- 

J240-3 

4596-3 

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259-1 

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4 

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Circumferences and Areas of Circles. 


109 


Circ. 

Dinme- r a 

ter. 


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1 - 
h~ 

2 - 

90- 

1 - 
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2 - 

91 — 

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92 

1 - 
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279- 9 

280- 3 

280- 7 

281- 1 

281-5 

281- 9 

282- 3 

282- 7 

283- 1 

283-5 

283- 9 

284- 3 

284- 7 

285- 1 

285-4 

285- 8 

286- 2 
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■! 288-6 
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■-289-8 
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290-5 

290- 9 

291- 3 
291-7 


Area. 


Circ. 

Aree. 


Circ. 

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f§ 

Diame¬ 

ter. 

o 

jIIpp* 

Diame¬ 

ter. 

o 

m 

6221-1 

93 

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6792-9 

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r| 304-7 

7389-S 

6238-6 


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7408-8 

6256-1 

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-292-9 

6829-4 

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7427-9 

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L 294-1 

6884-5 

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r 294-5 

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307- 

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6361-7 

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307-8 

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309-4 

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- 297-2 

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310-2 

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7893-3 

6701-9 


-302-7 

7294-9 


315-3 

7913-1 

6720-0 

i- 

-1303-1 

7313-8 

i 

315-7 

7932-7 

6738-2 


J303-5 

7332-8 


316-0 

7942-4 

6756-4 

$ - 

303-9 

7351-7 

1 -- 

316-4 

7972-2 

6776-4 

304-3 

7370-7 


316-8 

7991-9 


WXPL A 1 S ATION OF THE TABLE FOR SEGMENTS, &c. 

The chord divided by the height is the gauge in the Table, the quotient in the 
first column. 

k = tabular coefficient, always to be multiplied by the chord. 

To find the angle of an arc of a circle'. 

RULE. Divide the base (chord) of the arc by its height, (sine verse ) and find 
the quotient in the first column. The corresponding number in the ■■-econd 
column is the angle of the arc in degrees of the circle. 

To find the radius of an arc of a circle* 

RULE. Divide the chord of the arc by its height, and find the quotient in 
the first column. The corresponding number in the third oolumn, multiplied 
by the chord, is the radius of the arc. 




































110 Table for Segments &c., of a Circle. 


Chord div. 
by height. 

] Centre 

Angle v. 

1 Radius 
r = ft c. 

Cir. Arc. 

1 = k c- 

Area Seg. 

0 = ft C3. 

Surface 
a = ft c a 

Solidity 

C— k ; 3 

-■ - 1 

Chord 
e = ft r. 



’<5> 






\y 

'v'' 

t ;; 


\ / 
V 



458-08 

1 

57-296 

1-0000 

•01091 

•78539 

•00085 

•01744 

229-18 

2 

28-649 

1-0000 

•00218 

•78549 

•00172 

•03490 

152-77 

3 

19-101 

1-0000 

•00327 

•78462 

•00255 

•05234 

114-57 

4 

14-327 

1-0000 

•00436 

•78574 

•00310 

•06978 

84-747 

5 

11-462 

1-0001 

.00647 

•78586 

•00401 

•08722 

76-375 

6 

9-5530 

1-0003 

•00741 

•78599 

•00514 

•10466 

65-943 

7 

8-1902 

1-0004 

•00910 

•78621 

•00592 

•12208 

57-273 

8 

7-1678 

1-0006 

•01089 

•78630 

•00686 

•13950 

50-902 

9 

6-3728 

1-0008 

•01254 

•78665 

•00772 

•15690 

45-807 

10 

5-7368 

1-0011 

•01407 

•78695 

•00857 

•17430 

41-203 

11 

5-2167 

1-0013 

•01552 

•78730 

•00964 

•19168 

38-133 

12 

4-7834 

1-0016 

•01695 

•78725 

•01031 

•20904 

35-221 

13 

4-4168 

1-0019 

•01841 

•78794 

•01114 

•22640 

82-742 

14 

4-1027 

1-0023 

•02000 

•78832 

•01199 

•24372 

30-514 

15 

3-8307 

1-0027 

•02157 

•78889 

•01288 

•26.04 

28-601 

16 

3-5927 

1-0029 

•02269 

•78909 

•01375 

•27834 

26-915 

17 

3-3827 

1-0034 

•02434 

•78969 

•01462 

•29560 

25-412 

18 

3-1962 

1-0039 

•02592 

•79028 

•01542 

•31286 

24-068 

19 

3-0293 

1-0044 

•02744 

•79084 

•01635 

•33008 

22-860 

20 

2-8793 

1-0048 

•02878 

•79140 

•01722 

•34728 

21-760 

21 

2-7440 

1-0054 

•03040 

•79234 

•01802 

•36446 

20-777 

22 

2-6222 

1-0059 

•03178 

•79300 

•01897 

•38160 

19-862 

23 

2-5080 

1-0066 

•03343 

•79340 

•01984 

•39S72 

19-028 

• 24 

2-4050 

1-0072 

•03493 

•79416 

•02072 

•41582 

18*261 

25 

2-3101 

1-0078 

•03639 

•79486 

•02159 

•43286 

17-553 

26 

2-2233 

1-0084 

•03784 

•79530 

•02248 

*44990 

16-970 

27 

2-1418 

1-0091 

•03970 

•79639 

•02315 

•466S8 

16-288 

28 

2-0673 

1-0101 

•04115 

•79748 

•02424 

•48384 

15-721 

29 

1-9969 

1-0105 

•04230 

•79811 

•02511 

•50076 

15-191 

-14-970 

30 

31 

1-9319 

1-8710 

1-0113 

.1-0121 

•04385 

•04476 

•79907 

•78530 

•02600 

•02692 

•51762 

'53446 

14-230 

32 

1-8140 

1-0129 

•04710 

•80098 

•02778 

•55126 

13-796 

33 

1-7605 

1-0138 

•04842 

•80181 

•02866 

•56802 

13-382 

34 

1-7102 

1-0146 

•049S9 

•80300 

•02956 

.58479 

12-994 

35 

1-6-628 

1-0155 

•05137 

•80405 

•03046 

•60140 

12-733 

36 

1-6184 

1-0167 

•05311 

•80531 

•03137 

•61802 

12-473 

37 

1-5758 

1-0174 

•05401 

•80622 

•03226 

•63460 

11-931 

38 

1-5358 

1-0184 

•05628 

•80713 

•03328 

•65112 

11-621 

39 

1-4979 

1-0194 

•05755 

•80850 

•03418 

•66760 

11-342 

40 

1-4619 

1-0204 

•05899 

•80987 

•03506 

•68404 

11-060 

41 • 

1-4266 

1-0207 

•06001 

•81046 

•03589 

•70040 

10-791 

42 

1-3952 

1-0226 

•06196 

•81240 

•03680 

•71672 

10-534 

43 

1-3643 

1-0237 

•06359 

•81377 

•0.3773 

•73300 

10-28'9 

44 

1-3347 

1*0248 

•06574 

•81505 

•03S64 

•74920 

10-043 

45 

1-3066 

1-0260 

•06628 

•81756 

•03890 

•76536 

9-8303 

46 

1-2797 

1-0272 

•06826 

.81795 

•04050 

•78146 

9-6153 

47 

1-2539 

1-0290 

•06998 

•81939 

•04143 

•79748 

9-4092 

■J 

.u 

1-2289 

1-0297 

•09138 

•82064 

-04247 

•81346 

J 





































Table for Segments Ac., of a Circle. 


Ill 


Chord div. | 
by height. 

Centre 
Angle v. 

Rad iu9 
r = k c. 

Cir. Arc. 
b — k c. 

Area Seg. 
o = k c 2 . 

j Surface 

a ==: h c 2 . 

Solidity 

C = h J , 

Chord 
c — k r. 



<' 

FT— 

V ^ 





s' 

\c? 

n/ 

\ 




v- 

XT 7 


9-2113 

49 

1-2057 

1-0309 

•07290 

•82244 

•04330 

•82938 

9-0214 

50 

1-1831 

1-0323 

•07453 

•82384 

•04424 

•84522 

8-8387 

51 

1-1614 

1-0336 

•07611 

•82562 

•04519 

•86102 

8-6629 

52 

1-1406 

1-0349 

•07758 

•82729 

•04614 

•87674 

8-4462 

53 

1-1206 

1-0364 

•07959 

•83363 

•04685 

89238 

8-3306 

54 

1-1014 

1-0378 

•0S083 

•83072 

•04805 

•90798 

8-1733 

55 

1-0828 

1-0393 

•08246 

•83249 

•04901 

•92348 

8-0215 

56 

1-0650 

1-0407 

•08400 

•83422 

•05002 

•93894 

7-8750 

57 

1-0478 

1*0422 

•08579 

•83602 

•05098 

•95430 

7-7334 

58 

1-0313 

1-0431 

•08680 

•83796 

•05191 

•96960 

7-5895 

59 

1-0154 

1-0454 

•08S91 

•84064 

•05299 

•98484 

7"4565 

60 

1-0000 

1-0470 

•09106 

•84266 

•05400 

1-0000 

7-3358 

61 

•98515 

1-0486 

•09209 

•84380 

•054' 6 

1-0150 

7-2118 

62 

-97080 

1-0503 

•09375 

•84581 

•05583 

1-0300 

7-0914 

63 

•95694 

1-0520 

•09540 

•84791 

•05684 

1-0450 

6-9748 

64 

•94352 

1-0537 

•09697 

*84996 

•05784 

1-0598 

6-S616 

65 

•93058 

1-0555 

•09865 

•85215 

•05S85 

1-0746 

6-7512 

66 

•91804 

1-0573 

•10036 

•85441 

•059S7 

1-0892 

6*6453 

67 

•90590 

1*0591 

•10201 

•85640 

•06088 

1-1038 

6-5469 

68 

•89415 

1*0610 

•10367 

•85815 

•06181 

1-1184 

6-4902 

69 

•88276 

1-0629 

•10520 

•85464 

•06201 

1-1328 

6-3431 

70 

•87172 

1-0648 

•10710 

•86350 

•06396 

1*1471 

6-2400 

71 

•86102 

1-8668 

•10887 

•86699 

•06515 

1-1614 

6-1553 

72 

•85065 

1-0687 

•11046 

•86S34 

•06604 

1-1755 

6-0652 

73 

•84058 

1-0708 

•11225 

•870S1 

•06709 

1*1896 

5-9773 

74 

•83082 

1-0728 

•11385 

•87935 

•06815 

1-2036 

5-8918 

75 

•82134 

1-0749 

•11563 

•87590 

•06921 

1*2175 

5-8084 

76 

•81213 

1-0770 

•11736 

•87853 

•07037 

1-2313 

5-7271 

77 

■'•80319 

1-0792 

•11910 

•88120 

•07136 

1-2450 

5-6478 

78 

•79449 

1-0814 

•12072 

•88389 

•07244 

1-2586 

5-5704 

79 

•78606 

1-0836 

•12281 

•88677 

•07352 

1-2721 

5-.949 

80 

•77786 

1-0859 

•12441 

•88949 

•07462 

1-2855 

5-4254 

81 

•76988 

1-0882 

•12660 

•89161 

•07512 

1-2989 

5-3492 

82 

•76212 

1-0905 

•12793 

•89520 

•07683 

1-3121 

5-2705 

83 

•75458 

1-0920 

•12958 

•89958 

•07819 

1-3252 

5-2101 

-84 

•74724 

1-0953 

•13157 

*90095 

•07907 

1-3383 

5-1429 

85 

•74009 

1-0977 

•13330 

•90420 

•07960 

1-3512 

5-0772 

86 

•73314 

1-1012 

•13546 

90734 

•08102 

1-3639 

5-0134 

87 

-•72637 

1-1027 

•13704 

•91036 

•08440 

1-3767 

4-9501 

88 

•71978 

1-1054 

•13893 

•91363 

•00836 

1-3903 

4-8886 

89 

•71336 

1-1079 

•14078 

•91696 

•08450 

1-4818 

■ 4-8216 

90 

•70710 

1-1105 

•14279 

•92210 

•08621 

1-4142 

4-7694 

91 

•70101 

1-1132 

•14449 

•92352 

•0S716 

1-4265 

4-7117 

92 

•69508 

1-1159 

•14643 

•92476 

•0S798 

1-4387 

4-6615 

93 

•68930 

1-1186 

•14817 

•92914 

•0S932 

1-4507 

4-5999 

94 

•68366 

1-1211 

•15009 

•93385 

•09076 

1*4-627 

4-5453 

95 

•67817 

1-1242 

•15211 

•93746 

•09197 

1-4745 

4-4845 

-- 

96 

•67282 

1-1271 

•15375 

•94272 

•09348 

1*4863 



































112 


Table roa Sfoment.s Ac., of \ Circle. 


Chord div. 

Centre 

Radius 

Cir. Arc. 

by height. 

Angle v. 

r = 

k c. 

b = k c. 



/ 

r y 

s' 


\ * s 

V' 

\ 


4-4398 

97 

•66760 

1-1300 

4-3859 

98 

•66250 

1-1329 

4*3383 

99 

•65754 

M359 

4-2862 

100 

•65270 

1-1382 

4-2406 

101 

•6479S 

1-1420 

4-1930 

102 

•64338 

1-1451 

4-1570 

103 

•63 

889 

1-1483 

4-1006 

104 

•63450 

1-1515 

4-0555 

105 

•63023 

1-1547 

4*0113 

106 

•62607 

1-1580 

3-9679 

107 

•62200 

1-1614 

3-9252 

108 

•61803 

1*1648 

3-8832 

109 

•61416 

1-1682 

3-8419 

iio 

•61039 

1-1716 

3-8013 

in 

•60670 

1-1752 

3-7612 

112 

•60325 

1-1790 

3-7221 

113 

•59960 

1-1823 

3*6837 

114 

•59618 

1-1859 

3-6454 

115 

•59284 

1*1897 

3-6086 

116 

•58959 

1-1934 

3-5712 

117 

•58641 

1-1972 

3-5349 

118 

•58331 

1*2011 

3-4992 

119 

*58030 

1*2050 

3-4641 

120 

*57735 

1-2089 

3-4296 

121 

•57450 

1*2130 

3-3953 

122 

•57168 

1-2177 

3-3616 

123 

•56895 

1*2213 

3-3285 

124 

•56628 

1-2253 

3-2940 

125 

•56370 

1-2295 

3-2637 

126 

•56116 

1-2338 

3-2319 

127 

•55870 

1-2381 

3-2006 

128 

•55630 

1-2425 

3-1716 

129 

•55396 

1-2470 

3-1393 

130 

•55169 

1-2515 

3-1093 

131 

-54947 

1-2561 

3-0805 

132 

•54732 

1-2607 

3-0555 

133 

•54522 

1-2654 

3-0216 

134 

•54318 

1-2701 

2-9777 

135 

•54120 

1-2749 

2-9651 

136 

•53927 

1-2798 

2-9374 

137 

•53740 

1-2847 

2-9115 

138 

•53557 

1-2897 

2-8829 

139 

•53380 

1-2948 

2-8562 

140 

•53209 

1-2999 

2-8299 

141 

•53042 

1-3051 

2-8038 

142 

•52881 

1-3065 

2*7781 

143 

•52724 

1-3157 

2-7527 

144 

•52573 

1*3211 


I 


Area Seg. 

Surface 

Solidity 

Chord 

o = k rA 

c«’ 

o 

II 

e 

C — 4 c*. 

e = h r. 





s 


NX 


•15600 

•94470 

•09442 

1-4979 

•15801 

•94852 

•09567 

1*5094 

•15995 

•9523.6 

•09693 

1*5208 

•16180 

•95682 

•09831 

1*5321 

.16393 

•96011 

•09856 

1-5432 

•16610 

•96412 

•10076 

1-5543 

•16925 

•96568 

•10557 

1*5652 

•17001 

•97246 

•10273 

1-5760 

•17204 

•97643 

•10471 

1-5S67 

•17414 

•98067 

•10601 

1-5973 

•17619 

•98495 

•10735 

1-6077 

•17832 

•98931 

•10870 

1*6180 

•18041 

•99376 

•11007 

1-6282 

•18257 

•98827 

•11149 

1-638S 

•18472 

1-0028 

•11284 

1-6482 

•18696 

1-0077 

•11426 

1-6581 

•18900 

1-0122 

•11566 

1-6677 

•19117 

1-0169 

•11709 

1-6773 

•19339 

1-0218 

•11853 

1-6867 

•19559 

1-0266 

•11995 

1-6961 

•19787 

1-0317 

•12145 , 

1-7053 

•20009 

1-0368 

•12294 

1-7143 

*20227 

1-0417 

•12444 

1-7232 

*20453 

1-0472 

•12596 

1-7320 

•20678 

1-0525 

•12748 

1-7407 

*20945 

1-0578 

•12903 

1*7492 

•21175 

1*0634 

•13060 

1-7576 

•21399 

1*0690 

•13218 

1-7659 

•21538 

1-0753 

•13391 

1-7740 

•21859 

1*0803 

•13558 

1-7820 

•22121 

1-0862 

•13701 

1-7898 

*22370 

1*0921 

•13866 

1-7976 

•22617 

1*0974 

•14028 

1-8051 

*22865 

1*1040 

*14202 

1-S126 

*23113 

1*1104 

•14371 

1-8199 

•23372 

1-1164 

•14537 

1-8271 

•23603 

1-1212 

•14676 

1-8341 

*23892 

1-1295 

•14894 

1-8410 

•24198 

1-1420 

•15209 

1-8477 

•24364 

1-1428 

•15252 

1-8543 

•24676 

1*1495 

•15422 

1-8608 

•24938 

1-1558 

•15605 

1-8671 

•25222 

1*1634 

•15S07 

1-8733 

•25485 

1-1705 

•15996 

1-8794 

•25759 

1-1777 

•16201 

1-8853 

•25936 

1-1851 

•16381 

1-8910 

•26320 

1-1925 

•16577 

1-8966 

•26604 

1-2000 

•16776 

1-9021 


































Table for Segments &c., of a Circle. 113 


Chord div. 
by height. 

Centre 
Angle t>. 

Radius 
r = ft c. 

Cir. Arc. 
i = ic. 

Area Seg. 
a = ft c 3 . 

Surface 

1 = ft c 3 

Solidity 

C — k c*. 

Chord 
c — A r. 




~~v. 

N 






\ * s* 

V' 


N 


\ v /' 


'/ 



2*7276 

145 

*52426 

1*3265 

•26889 

1*2077 

*16965 

1*9074 

2*7002 

146 

*52.2^4 

1-3320 

•27196 

1*2166 

*17209 

1*9126 

2*6816 

147 

*52147 

1*3377 

•27449 

1-2219 

•17205 

1*9176 

2*6533 

148 

*52015 

1*3433 

•27772 

1-2318 

•17605 

1*9225 

2*6301 

149 

*51887 

1*3491 

•28168 

1*2396 

•17809 

1-9272 

2*6064 

150 

*51764 

1-3549 

•28369 

1*2476 

‘18023 

1*9318 

2*5830 

151 

*54645 

1*3608 

•28,674 

1*2563 

•18666 

1*9363 

2*5598 

152 

*51530 

1*3668 

•28983 

1*2648 

: 18751 

1*9406 

2*5239 

153 

*51420 

1*3729 

•29397 

1-2891 

; 18845 

1*9447 

2*5143 

154 

•51315 

1-3790 

•29607 

1*2824 

*18913 

1*9487 

2*4919 

155 

•51214 

1-3852 

•29928 

1-2914 

•19147 

1-9526 

2*4699 

156 

•51117 

1*3919 

•30259 

1-3004 

•19374 

1*9563 

2*4478 

157 

•51014 

1*3973 

•30560 

1-3094 

•19607 

1*9598 

2*4262 

158 

•50936 

1*4043 

•30905 

1*3191 

•20029 

1*9632 

2*4047 

159 

•50851 

1*4109 

•31239 

1*3287 

•29095 

1*9663 

2*3835 

160 

•50771 

1-4175 

•31575 

1*3368 

‘20342 

1-9696 

2*3613 

161 

•50695 

1*4243 

•31931 

1*3490 

•20609 

1-9725 

2*3417 

162 

•50623 

1*4311 

•32263 

1-3583 

*20847 

1*9753 

2*3211 

163 

•50555 

1*4380 

•32618 

1-3682 

•21105 

1*9780 

2*3004 

164 

•50491 

1*4450 

•32969 

1*3791 

*21371 

1*9805 

2*2805 

165 

•50431 

1-4520 

•33327 

1*3895 

*21634 

1*9829 

2*2605 

166 

•50374 

1*4592 

•33684 

1*4021 

•21904 

1*9851 

2*2408 

167 

•50323 

1*4665 

•34048 

1*4111 

•22177 

1*9871 

2*2212 

168 

•50275 

1*4739 

•34422 

1*4222 

•21946 

1*9890 

2*2013 

169 

•50231 

1*4813 

•34802 

1*4344 

•22766 

1*9908 

2*1826 

170 

•50191 

1-4889 

•35230 

1*4476 

•23028 

1*9924 

2*1636 

171 

•50154 

1*4966 

•35563 

1-4565 

•23266 

1*9938 

2*1447 

172 

•50122 

1*5044 

•35953 

1-4684 

•23650 

1*9951 

2*1271 

173 

•50093 

1*5123 

•36337 

1-4797 

•23900 

1*9962 

2*1075 

174 

•50068 

1*5202 

•36747 

1-4927 

•24225 

1*9972 

2*0892 

175 

•50047 

1*5283 

•37152 

1-5052 

•24537 

1*9981 

2*0710 

176 

•50030 

1*5365 

•37562 

1-5179 

•24856 

1*9988 

2*0530 

177 

•50017 

1-5448 

•37974 

1-5308 

•25179 

1*9993 

2*0352 

178 

•50007 

1*5533 

•38401 

1*5439 

•25531 

1*9996 

2*0175 

179 

•50002 

1-5618 

•38828 

1*5573 

•25840 

1*9999 

2*0000 

180 

•50000 

1*5707 

•39269 

1*5708 

•26179 

2*0000 


To find the length of an arc of a circle* 

RULE. Divide the chord of the arc by its height, and find the quotient in 
the first column. The corresponding number in the fourth eolumn multiplied 
by the chord is the length of the arc. 


To find the area of a segment of a circle* 

RULE. Divide the chord of the segment by its height, and find the quotient 
in the first column. The corresponding number in the fifth column multiplied 
by the square of the chord, is the area of the segment. 
































114 


O EFFICTF.NT FOR CAPACITY AND WEIGHT. 


Coefficient for Capacity and Weight, 


Names of Substances. 

Cubic inches, - 
Cubic feet, - - 
Gallons, - - - 
Water, fresh, - 
Water, salt, - - 
Oil, - - - - - 

Cast-iron, - - 
Wrought-iron, - 
Steel, - - - - 
Brass, - - - - 
fin, ... - 
Lead, ? - - * 
Zinc, - - - - 
Copper, - - - 
Mercury,. - - 
Stone, common, 
Clay, - • - - 
Earth, compact, 
Earth, loose, - 
Oak, dry, - - 
Bine, - - - - 
Mahogany, - - 
Coal, stone, - - 
C.iarcoal, - - - 


gggggg 


■ / \ 

-;- 


\ 

d 



FFE. 

Fa. 

Hi. 

FFr 

Fi*. 

it*. 

F*. 

i 3 . 

1728 

12 

i 

1356 

9-42 

0-78’ 

903-7 

0-523 

1 

-..694 

•.58 

0-785 

-..549 

•.44 

0-523 

•.3 

7-476 

0-052 

-...433 

5-868 

• .408 

*.34 

3-91 

-..226 

62‘5 

0-433 

0-036 

49 

0-34 

•. 283 

32-7 

0-019 

64-3 

0-445 

0-037 

' 50-4 

0"35 

0-029 

33-6 

0-02 

57-5 

0-4 

0-033 

45-1 

0-313 

0-026 

30 

0-017 

450 

312 

0-26 

353 

2-45 

0-204 

235 

0-136 

487 

3-37 

0-281 

382 

2-65 

0-221 

255 

0-147 

490 

3-4 

0-283 

385 

2-67 

0-222 

257 

0-149 

532 

3-68 

0-307 

417 

2-9 

0-241 

278 

0-161 

456 

3-16 

0-263 

358 

2-48 

0-207 

239 

0-138 

710 

4-02 

0-41 

557 

3-87 

0-322 

371 

0-215 

440 

3-05 

0-254 

345 

2-4 

0-2 

230 

0-133 

556 

3-85 

0-321 

436 

3-03 

0-252 

291 

0-168 

850 

59 

0-491 

666 

4-63 

0-385 

445 

0-257 

156 

1-08 

0-09 

122 

0-85 

0-071 

82 

0-047 

135 

0-936 

0-078 

106 

0-735 

0061 

70 

0-04 

127 

0-88 

0-0733 

99 

0-692 

0-058 

66 

0-038 

95 

0-66 

0-055 

74 

0-517 

0-043 

50 

0-02!' 

58 

0-4 

0-033 

44 

0-316 

0-026 

30 

0-017 

30 

0-208 

0-017 

24 

0-163 

0-014 

16 

0-009 

66 

0-457 

0-038 

52 

0-3.6 

0-03 

34 

0-02 

54 

0-375 

0-031 

42 

0-294 

0-024 

28-2 

0-016 

27-5 

0-19 

0-016 

21 

0-15 

O-012 

H-4 

0-008 


To Find the Weight and Capacity l>y this Tahle 

RULE. The product of the dimensions in feet or in inches, as noted in the 
columns, multiplied by the tabular coefficient, is the capacity of the solid, or 
weight in pounds avoirdupois. 

Example 1. A cistern is 6 feet long, 27 inches wide, and 20 inches deep. 
How many gallons of liquid can be contained in it ? 

6X 27 X 20X0-052 = 168 48 gallons. 

Example 2. A cast-iron cylinder is 4-5 feet long, and 7‘5 inches diameter. 
Required the weight of it? 

4-5+7 , 5‘ 2 X2-45 = 620 pounds. 














































Tat)le of 8lli Ordinates, for Railroad Curves* 115 


4ni/le. 


Ordinates . 


Angie. 


Ordinates . 


VV 

1. 7. 

2. 6. 

3. 5. 

4. lx. 

w 

1. 7. 

2. 6. 

3. 5. 

4. h. 

1° 

•00084 

• 00164 . *00193 

•00218 

5 3° 

•05313 

•08932 

•11063 

•11773 

2, 

•00191 

•00327 

•00409 

•00436 

54 

•05422 

•09130 

•11318 

•12003 

3 

•00299 

•00522 

•00561 

*00659 

55 

05531 

•09308 

•11510 

•12235 

4 

•00382 

•00654 

•00818 

•00872 

56 

•05646 

•09487 

•11731 

•12466 

5 

•00437 

•00818 

•01023 

•01091 

57 

•05760 

•09673 

•11950 

•12698 

6 

•00573 

•00928 

•01228 

•01309 

58 

•05875 

*09853 

•12170 

•12932 

7 

•00675 

•01173 

•01432 

-01527 

5 9 

•05989 

•10037 

•12393 

•13162 

8 

•00764 

•01309 

•01639 

•01746 

GO 

•06094 

•10220 

*12612 

•13397 

9 

• 0 0 845 

•01474 

•01842 

•01964 

6 1 

•06261 

•10427 

•12840 

•13631 

1 0 

•00955 

•01637 

•02047 

•02183 

62 

•06331 

•10593 

•13054 

•13866 

1 1 

•01053 

•01801 

•02250 

•02402 

63 

•06451 

•10781 

•13281 

•14101 

12 

•01146 

•01965 

•02456 

•02620 

64 

•06570 

•10964 

*13505 

•14337 

13 

•01245 

•02129 

02662 

•02839 

65 

•06681 

•11101 

•13765 

•14573 

14 

•01284 

•02271 

•02861 

•03058 

66 

•06805 

•11342 

*13956 

•14810 

15 

•01438 

•02461 

•03081 

•03282 

67 

•06914 

•11532 

• 141 S 1 

•15048 

16 

•01535 

■02625 

•03277 

•03496 

68 

•07040 

•11721 

•14409 

•15286 

1 7 

•01630 

•02789 

•03484 

•03715 

69 

•07168 

•11912 

•14637 

•15526 

1 8 

•01730 

•02956 

•03693 

•03935 

70 

•07284 

•12103 

*14864 

•15765 

1 9 

•01858 

•03125 

•03996 

•04154 

7 1 

•07407 

•12294 

•15087 

•16005 

20 

•01922 

•03286 

•04103 

• 0437 4 

72 

•07535 

•12485 

•15323 

•16245 

21 

•02022 

•03453 

•04309 

•04594 

73 

•07656 

•12685 

•15555 

•16487 

2 2 

•02119 

•03619 

•04522 

•04814 

74 

•07784 

•12877 

*15785 

•16729 

23 

•02215 

•03787 

•04720 

•05034 

75 

•07912 

•13078 

•16016 

•16972 

24 

•02311 

•03934 

•04930 

•05255 

76 

•08040 

•13292 

*16247 

•17216 

25 

•02413 

•04117 

•05138 

•05475 

77 

•08168 

•13472 

*16482 

•17460 

26 

•02508 

•04283 

•05346 

•05696 

78 

•08297 

•13670 

•16716 

•17706 

2 7 

•02610 

■04457 

•05552 

•05917 

79 

•08426 

•13868 

•16951 

•17951 

28 

•02708 

•04621 

•05761 

•06139 

80 

•08560 

•14070 

•17187 

•18198 

29 

•02813 

•04793 

•05970 

•06361 

81 

■08695 

•14274 

•17423 

•18445 

30 

•02911 

•04970 

•06188 

•06582 

8 2 

•08829 

•14477 

•17660 

•18694 

31 

•03005 

•05125 

06386 

•06804 

83 

•08944 

•14681 

■17901 

•18943 

32 

•03107 

•05298 

•06596 

•07027 

84 

•09105 

•14888 

•18140 

•19193 

33 

•03191 

•05464 

•06806 

•07250 

85 

•09235 

•15120 

•18379 

•19444 

3 4 

•03310 

•05637 

•07016 

•07477 

86 

•09377 

•15304 

•18622 

•19695 

35 

•03412 

•05804 

•07424 

•07695 

87 

•09 •;* 

•15509 

•18865 

■19946 

36 

•03515 

•05992 

•07452 

•07919 

88 

•09660 

•15756 

•19108 

•20201 

37 

•03616 

•06147 

•07646 

•08143 

89 

•09780 

•15931 

•19350 

•20555 

38 

•03718 

•06327 

•07858 

•08367 

90 

•09944 

•16144 

•19597 

•20710 

39 

03821 

•06492 

•08069 

•08591 

91 

•10098 

•16359 

•19842 

•20966 

4 0 

•03905 

•06631 

•08243 

•08816 

92 

•10240 

•16575 

•20092 

•21223 

41 

•04030 

•06836 

•08494 

•09041 

93 

•10384 

•16787 

•20338 

•21481 

4 2 

•04133 

•07012 

•08707 

•09266 

94 

•10537 

•17005 

•20589 

•21740 

43 

•04241 

•07182 

•08920 

•09492 

95 

•10692 

•17224 

• 20 S 37 

•22000 

44 

•04363 

•07353 

•09130 

•09719 

96 

•10851 

•17444 

•21091 

•22262 

45 

•0 522 

•07531 

•09346 

•09945 

97 

•10997 

•17666 

•21342 

•22523 

46 

•04556 

•07706 

•09562 

•10172 

98 

•11150 

•17888 

•21596 

•22786 

47 

•04682 

•07894 

•09790 

•10400 

99 

•11310 

•18111 

•22800 

•23050 

4 8 

•04833 

•08059 

•09991 

•10627 

100 

•11468 

•18354 

•22107 

•23315 

4 9 

•04879 

•08236 

•00207 

•10856 

101 

•11626 

•18500 

•22364 

•23596 

5 0 

•01982 

•08413 

•00422 

•11085 

102 

•11791 

•18793 

•22623 

•23848 

5 1 

•05096 

•08593 

•10639 

•11314 

103 

•11959 

•19021 

•22876 

•24107 

52 

•05204 

•08768 

•10855 

•11543 

1 04 

•12116 

_ 

•19256 

•23147 

• 24386 j 






































116 


Rail Road Curves. 


RAIL ROAD CURVES. 

When Railroads are to be connected by curves, we commonly have given the 
distance (chord c,) between the two ends o o of the tracks, and the tangential 
angle v. By these the curve is to be constructed. 

Example 1. Fig. 128. The chord C = 168 feet, and the tangential angle 
v = 19° 30'. Required the centre angle w =, and the radius R = ? 

w = 2(19° 30') = 39°. R = 3 *k c = 1-4979X168 = 251-647 feet. 

k = See Table for Segments, &c., of a circle. 

By Tangential Angles. 

The curve to be laid out by the three tangential angles ror, ron, and noo, 
each angle = = 6° 30'. Required the chord r = ? 

The centre angle for the chord r is 

2X(6° 30') = 13°, and r — 13 k R = 0-2264X251-647 = 56-974 feet. 

By Angles of Deflexion. 

Divide the centre angle w into an even number of parts = z. Set off at o the 
angle z = r o n, and bisect it into ror and ron ,—find the chord r, and sub-chord 
a, and continue as shown by Figure. 

Example 2. Fig. 128. The tangential angle v — 78°, and the chord G = 638 
feet. Required the centre-angle w — ? Radius R = ? Chord r — ? and the sub¬ 
chord a = ? 

w = 2X78° = 156°. R = * k c = 0-51117X638 = 326-126 feet. 

Let the curve be laid out by 6 angles of deflexion, and z — 1X156° = 26°, and 
r = *«k R = 0-4499X326-126 == 146-73 feet. 
a = 2 6 k r = 0-4495x116-73 = 66-012 feet. 

By Ordinates. 

Example 3. Fig. 129. The chord C = 368 feet, and v — 35°. Required the 
height h — ? 

h = ^C( cosec.v — cot.u). 

From ------- cosec.36° — 1-70130 

Subtract ------- cot.36° = 1-37638 

The height h = 0-32492X184 = 59-785 feet. 

At x = 92 feet from h. Required the ordinate y ? 

2X92 sin.36° 


wn.z = 


y = JX 368 


( 


368 
cos.!7° 6' 


0-2938926 = sin.l7° 6'. 


’) 


sin.36° 

By Siil>-Cl»ofds 


cot.36° 1 = 45-9448 feet. 


Example 4. Fig. 130. The ends o and o of the tracks form different angles w 
and W to the chord C, and therefore must be connected by two curves of differ¬ 
ent radii, R a .d r. The chord C = 869 feet, w = 38°, and W = 86°. Required 
the distance from o to the height h, n —l sub-chord b = ? sub-chord a = ? 
radii R and r = ? 

v = 4X38° = 19°, and V = iX86° = 43°. 

869 tan.19° , 

n — -—--— = 234-35 feet. 

tau,19°-|-tan.43 0 

6 = 234-35 sec.43° = 320-42 feet. I R = ™ka — 1-5358X671-21 = 1030-2 ft. 
a = sec.l9°(869 — 234-35) = 671-21 ft. | r = 8S A b = 0-73314X320-42 = 234-91 ft. 

By Eiglit. Ordinates. 

Exanple 5. Fig. 133. Required 8 ordinates for a curve of chord C = 710 feet 
and the centre angle w = 69°? (See Table on the preceding page.) 

1st and 7th Ordinates 0-07168X710 = 50*8928 feet. 

2nd “ 6th “ 011912X710 = 84-5752 “ 

3rd “ 5th “ 0-14637X710 =- 103-9227 “ 

4th or height h 0-15526X710 = 110-2346 “ 









Railroad Curves. 


117 




L28, 

By angles of deflexion, 
w = 2v, R = w k C = 1 C cosec. r. 

r = z k R, a = z Jc r — 2r sin. £ 2 . 


129. 

By Ordinates. 
h= $C( cosec.v — eot.v). 

y^iC(^l-l—cot.v ), 
\ sin.v / 

2x sin.v 
sin. 2 =-—— . 


130. By Sub-chords. 
C tan.v 

y 


tan.y+tan. 
b = n sec.F, 
a = sec.vl C — n), 


h = n tan.F, 


w — 2v 
W = 2 V’ 


Hr & 


! 131 

Parallel tracks by a reverse curve. 

Formulas same as above. 

The length o o = 2c, length of 
a circle arc l = 0-035 c R. 


132. 



The 

greatest radius in a reverse 


curve. 

w = 

i(F+3y), 

w= w+V— V, 

a = w 

kR, b = w 

kR, 

u = q 

sec.w(sin.F— 

- j/siu. 2 F— cos. 2 u>). 




h 


of \ ! 

/S»l > 

1 

2 

i \o 


133. 

Curve by 8 Ordinates. 

The ordinates are calculated in the 
accompanying Table, the chord C — 1 or 
the unit. 

If the angle w is large, or there he some 
obstacle on the chord C, find the height h 
and lay out the curve by two or more sets 
of 8 ordinates. 











































118 


Excavation and Embankment. 


EXCAVATION AND EMBANKMENT. 

Example 1. The Road-way of an excavated channel is r = 15 feet, the depth 
D = 9 feet, and the breadth at the top b — 46£ feet. Require the slope S — ? 


Formula 6. 


46-5 — J5 

s = = 1,75 or u t0 V 


Example 2. The Road way is to be r = 15, D = 18, and the slope S = If, 
Require the breadth b = ? and the cross-section A = ? 


Formula 4. 
Formula 7. 


b = 2 X 18 X 1‘25 + 15 = 60 feet. 
A = — ^ 60 -f- 15 j= 675 square feet. 


Example 3. The Road-way is to be r — 16 feet, the slope S — 1£, and the depth 
D = 11 feet. Required the area of Cross-section A = '( 


Formula 9. 


A = 11 (11 Xlf +0 = 357 *5 square feet. 


Example 4. The Road-way r = 18 feet., slope S = If, d = 14 feet 6 inches, and 
*he length from o is X = 55 feet. Required the cubic contents c = ? 


/1 

Formula 11. c = 55 X 14*5^- 
by 27 = 444.28 cubic yards. 


— )= 11995*676 cubic feet, divided 


Example 5. The Road-way is r = 16 feet, slope S — If feet, F ==s 17*5, c? = 7*4 
and the length L = 100 feet. Required the cubic content C = ? 

Formula 12. € - l 00 [l|(^ 5 3 + 7 ' 4 * + 17 * X 7>4 -) + -f TiJ] 

= 44445 cubic feet, or 1645*4 cubic yards. 

The computation is executed thus. 


17-5 

7-4 


17-5 

7.4 


700 

1225 


129*50 

17*5* = 306*25 ) From table 
7’4 a = 54*76 ^ of Squares. 


24*9 

8 

199*2 


3)«o5i a“;» 6 |a D p e .«d d * 

199-2 ’ 

X 100 = 44445. cubic feet. 




















Excavation and Embankment. 


719 


134 



Letters in tile Formulas correspond wftli tile Figure. 




D /T 


iS = cot. v, 

1. 

A= “o(* + 0» 

7. 



. d , , . 

8. 

a = D S, 

2. 

a = + r), 

a = D cot. v, - 

3. 

A = D(D S r), 

9. 



a = d(d S + i*), 

10. 

b = 2 D S + r, 

4. 


I 

II 

5. 


11. 




s - b Z 

2 D’ 

6, 

+ l (D + $].• - 

12. 


Letters Denote, 

A and a = Cross-Sections in square feet, of the excavated channel or 
embankment. 

D and d -- depth in feet, of the Sections. 

r — width in feet of the Road-Way. 

b = Base in feet of the embankment, or top breadth of the channel. 

L — length in feet, between the two Sections A and a. 

I = length in feet, from the Section a to the point o where the ground is 
level with the road. 

C = cubic contents in feet, between A and a. 

c = cubic contents in feet, between a and o. 

(S' = slope of the sides. The slope is commonly given in proportions, thus: 

Slope = H to 1,” which means, that the side slopes 1£ feet horizontally for 1 
foot vertical. 

v =s angle of the slope. 






































120 


Trigonometry. 


To Reduce Indies and Fractions thereof to Decimals of a 
Foot) and vice versa* 

PLATE I. 

This is a common decimal scale on which the 12 inches of a foot are laid 
out. Any length of a foot expressed by inches and the common fractions, 
intersects its own value in decimals, and are read off as on a common diagonal 
scale with 10 to the base. 

Example 1. How much is 8^ inches in decimals of a foot ? Find inches 
on the rule , which will be found to intersect 6875 on the decimal scale; 
cr 8£ in. = 0-6875 feet. 

The first figure 6 is marked at the top and bottom of the scale, the second 
figure 8 is the eighth vertical (or nearly so) line from 6; the third figure 7 is the 
seventh horizontal line marked at the ends of the scale, and the fourth figure 5 
is the horizontal dotted line between 7 and 8. 

For sixteenths see the fine single rule. 

Example'2. How much is 0-526 feet in inches? 6 ^ the answer. 

To Reduce Vulgar Fractions into Decimals and vice versa. 

PLATE II. 

This is a similar arrangement to the preceding one. The vulgar fractions 
are laid out so that the denominators are marked at the ends of the scale 
and the nominators on the line that joins the given denominators; by 
this arrangement the nominator of the vulgar fraction intersects its own value 
in decimals on the scale. To facilitate the operations it is best to imagine in 
which quarter the vulgar fraction is; as Is in the first, f in the second; f in 
the third, and qf in the fourth quarter, &c., &c.; each rule occupies a quarter 
of the scale, on which the vulgar fraction is to be found accordingly. 

Example 1. How much is in decimals ? 

On the third rule it will be found at 5625 of decimals, or = 0-5625. The 
first and second figures 56 are marked as described for Plate I, and the third, 
fourth, &c., &c., figures are written down on the horizontal line of the given 
fraction. 

Example 2. What nearest vulgar fraction answers to the decimals 0-39583 ? 

> the answer. 

These two diagrams are exceedingly useful in practice. By Plate II, Vulgar 
Fractions can be added and subtracted. 




TRIGONOMETRY. 

Trigonometry is that part of Geometry which treats of Triangles. It is di¬ 
vided into two parts, viz.: plane and spherical. 

Plane Trigonometry treats of triangles which are drawn (or imagined to.be) on 
a plane. Spherical Trigonometry treats of the triangles which are drawn (or im¬ 
agined to be) on a sphere. 

A triangle contains seven quantities, namely, three sides, three angles, and 
the surface; when any three of these quantities are given, the four remaining 
ones can by them be ascertained, (one side or the area must be one of the given 
quantities) and the operation is called solving the triangle , which is only an ap¬ 
plication of arithmetic on Geometrical objects. 

For the foundation of the above mentioned solution, there are assumed eight 
help quantities which are called Trigonometrical functions, and are here denoted 
with their names and number, corresponding with Figure 1 ? In the accompa¬ 
nying Tables, the functions are calculated at every 10 minutes per degree in the 
quadrant of the circle represented by Fig. 1 ? The angle for which the functions 
are mentioned, is the opening between the two lines 7 and 2, 3, this angle is de¬ 
noted by the letter C, and the expression sin.C. means the line 1 compared with 
the radius r as a unit. 








I ' 





































- 



To /educe Inches lo /or/. 


Plate I. 






































































































































To reduce vulgar fmctwjis to decimals 




n'ate/7. 



>, ^ H. Ci ^ 


Jli.. 1 ystrom. 














































































































































































































































































































































































































































































































































































ft 








• M ‘ 

















! v • ' J .. ' ; ' 





















~ 
















■* • 
















Trigonometry. 


121 


135 



1 

Sinus 

abbreviated 

sin.C. 

2 

Cosinus 


cos.C. 

8 

Sinus-versus 

it 

sinv.C. 

4 

Cosinus-versus 

<» 

cosv.C. 

5 

Tangent 

66 

tan.C. 

6 

Cotangent 

66 

cot.C. 

7 

Secant 

66 

sec.C. 

8 

Cosecant 

66 

cosec.C. 


r = Radius of the circle, which is the unit by which the functions are mea¬ 
sured. 



sin. 2 C+cos. 2 C. 


sec .C = 


1 

cos. C ’ 


tan.C 


sin.C 
cos .C’ 


cosec.C ■= 


1 

sin. C ’ 


tan. C 


1 _ 

cot . C’ 


sinv.C =1 — cos .C, 
cosv.C = 1 sin.C, 


cot.C 


cos. C. 
sin. C ’ 


cot.C 


1 

tan.C’ 


sin.2C = 2 sin.C cos.C, 

sin.^C = i/5sin. 2 C+sinv. 2 C), 

sin.(C+2?) = sin.C cos. 5+ 
sin.Z?cos.C. 


Positive and, Negative Signs. 


Angles. 

sin. 

cos. 

sinv. 

CO . 

tan. 

cot. 

sec. 

cosec. 

+0° 

+0 

+1 

+0 

+1 

+0 

+ 00 

+1 

+ 00 

-r-90 0 

+1 

+0 

+1 

+0 

+00 

+ 0 

+00 

+ 1 

+18,° 

±0 

-1 

+2 

+1 

+o 

Too 

—1 

+00 

+27G° 

—1 

+o 

+1 

+2 

+oo 

+' 

~oc 

—1 

4-360° 

+ 

+1 

+0 

+1 

+0 

— oo 

+1 

—oo 


When a Quantity has reached 0 or , it has ceased to exist, because it can 
not be increased or diminished. 

Example. What is the length of the secant for an angle of 74° 18'? 

Secant C = ^ = 3-695. 


ii 




































122 


Trigonometry. 


Natural Sine* 


Dog. 

*i- 

O' 

10' 

20' 

30' 

O 

50' 

60' 

T 

0 

•oocoo 

•00291 

•00581 

•00872 

•01163 

*01454 

•01745 

89 

1 

•01745 

•02036 

•02326 

•02617 

•02908 

•03199 

•03489 

88 

2 

•03489 

•03780 

•04071 

•04361 

•04652 

•04943 

•05233 

87 

3 

•05233 

•05524 

•05814 

•06104 

•06395 

•06685 

•06975 

86 

4 

•06975 

•07265 

•07555 

•07845 

•08135 

•08425 

•08715 

85 

5 

•08715 

•09005 

•09294 

•09584 

•09874 

•10163 

•10452 

84 

6 

•10452 

•10742 

•11031 

•11320 

•11609 

•11898 

•12186 

83 

7 

•12186 

•12475 

•12764 

•13052 

•13340 

•13629 

•13917 

82 

8 

•13917 

•14205 

•14493 

•14780 

•15068 

•15356 

•15643 

81 

9 

•15643 

•15930 

•16217 

•16504 

•16791 

•17078 

•17364 

80 

10 

•17364 

•17651 

•17937 

•18223 

•18509 

•18795 

•19080 

79 

11 

•19080 

•19366 

•19651 

•19936 

•20221 

•20506 

•20791 

78 

12 

•20791 

•21075 

•21359 

•21643 

•21927 

•22211 

'22495 

77 

13 

•22495 

•22778 

•23061 

•23344 

•23627 

•23909 

•24192 

76 

14 

•24192 

•24474 

•24756 

•25038 

•25319 

•25600 

•25881 

75 

15 

•25881 

•26162 

•26443 

•26723 

•27004 

•27284 

•27563 

74 

16 

•27563 

•27843 

•28122 

•28401 

•28680 

•289-58 

•29237 

73 

17 

•29237 

•29515 

•29793 

•30070 

•30347 

•30624 

•3090] 

72 

18 

•30901 

•31178 

•31454 

•31730 

•32006 

•32281 

•32556 

71 

19 

•32556 

•32831 

•33106 

•33380 

•33654 

•33928 

•34202 

70 

20 

•34202 

•34475 

•34748 

•35020 

•35293 

•35565 

•35836 

69 

21 

•35836 

•36108 

•36379 

•36650 

•36920 

•37190 

•37460 

68 

22 

•37460 

•37730 

•37999 

•38268 

•38536 

•38805 

•39073 

67 

23 

•39073 

•39340 

•39607 

•39874 

•40141 

•40407 

•40673 

66 

24 

•40673 

•40939 

•41204 

•41469 

•41733 

■41998 

•42261 

65 

25 

•42261 

•42525 

•42788 

•43051 

•43313 

•43575 

•43837 

64 

26 

•43837 

•44098 

•44359 

•44619 

•44879 

•45139 

•45399 

63 

27 

•45399 

•45658 

•45916 

•46174 

•46432 

•46690 

•46947 

62 

2S 

•46947 

•47203 

•47460 

•47715 

•47971 

•48226 

•484S0 

61 

29 

•48480 

•48735 

•48988 

•49242 

•49495 

•49747 

•50000 

60 

30 

•50000 

•50251 

•50502 

'50753 

•51004 

•51254 

•51503 

59 

31 

•51503 

•51752 

•52001 

•52249 

•52497 

•52745 

•52991 

58 

32 

•52991 

•53238 

•53484 

•53729 

•53975 

•54219 

•54463 

57 

33 

•54463 

•54707 

•54950 

•55193 

•55436 

•55677 

•55919 

56 

34 

•55919 

•56160 

•56400 

•56640 

•568S0 

•57119 

•57357 

55 

35 

•57357 

•57595 

•57833 

•58070 

•58306 

•58542 

•58778 

54 

36 

•58778 

•59013 

•59248 

•59482 

•59715 

•59948 

•60181 

53 

37 

•60181 

•60413 

•60645 

•60876 

•61106 

•61336 

•61566 

52 

38 

•61566 

•61795 

•62023 

•62251 

•62478 

•62705 

•62932 

51 

39 

•62932 

•63157 

•63383 

•63607 

•63832 

•64055 

•6427S 

50 

40 

•64278 

•64501 

•64723 

•64944 

•65165 

•65386 

■65605 

49 

41 

•65605 

•65825 

•66043 

•66262 

•66479 

•66696 

•669131 

48 

42 

•66913 

•67128 

•67344 

•67559 

•67773 

•67986 

•6S199j 

47 

43 

•68199 

•68412 

.68642 

•68835 

•69046 

*69256 

•694651 

; 46 

44 

•69465 

•69674 

•69S83 

•70090 

•70298 

•70504 

•70710 

45 


60' 

50' 

40' 

| 

30' 

20' 

10' 

0' 

Deg. 

9 


Natural Cosine. 




































Trigonometry. 


123 


!—-- 

I 

Natural Sine. 


1 

D<3 . 

0' 

1 10' 

20' 

30' 

40' 

50' 

| 60' 

1 

45 

•70710 

•70916 

•71120 

•71325 

•71523 

•71731 

•71933 

| 44 

46 

•71933 

•72135 

•72336 

•72537 

•72737 

•72936 

•73135 

1 43 

17 

•73135 

•73333 

•73530 

•73727 

•73923 

•74119 

•74314 

1 42 

48 

•74314 

•74508 

•74702 

•74895 

•75088 

•75279 

•75470 

1 41 

49 

•75470 

•75661 

•75851 

•76040 

•76229 

•76417 

•76604 

j 40 

50 

•76604 

•76791 

•76977 

.77162 

•77347 

•77531 

•77714 

1 39 

51 

•77714 

•77897 

•78079 

•78260 

•78441 

•78621 

•78801 

38 

52 

•78801 

•78979 

•79157 

•79335 

•79512 

•79688 

•79863 

37 

53 

•79863 

•8003S 

•80212 

•80385 

•80558 

•80730 

•80901 

36 

54 

•80901 

•81072 

•81242 

•81411 

•81580 

•81748 

•81915 

35 

55 

•81915 

•82081 

•82247 

•82412 

•82577 

•82740 

•82903 

34 

56 

•82903 

•83066 

•83227 

•83388 

•83548 

•83708 

•83867 

33 

57 

•83S67 

•84025 

•84182 

•84339 

•84495 

•84650 

•84804 

32 

58 

•84804 

•84958 

•85111 

•85264 

•85415 

•85566 

•85716 

31 

59 

•85716 

•85866 

•86014 

•S6162 

•86310 

•86456 

•86602 

30 

60 

•86602 

•86747 

•86891 

•87035 

•87178 

•87520 

•87461 

29 

61 

•87461 

•87602 

•87742 

•S788] 

•88020 

•88157 

•88294 

28 

62 

•88294 

•88430 

•88566 

•88701 

•88835 

•88968 

•89100 

27 

63 

•89100 

•89232 

•89363 

•89493 

•89622 

•89751 

•89879 

26 

04 

•89879 

•90006 

•90132 

•90258 

•90383 

•90507 

•90630 

25 

65 

•90630 

•90753 

•90875 

•90996 

•91116 

•91235 

•91354 

24 

66 

*91354 

•91472 

•91589 

•91706 

•91811 

•91936 

•92050 

23 

67 

•92050 

•92163 

•92276 

•92387 

•92498 

•92609 

•92718 

22 

68 

•92718 

•92826 

•92934 

•93041 

•93147 

•93253 

•93358 

21 

69 

•93358 

•93461 

•93564 

•93667 

•93768 

•93869 

•93969 

20 

70 

•93 969 

•9 1068 

•94166 

•94264 

•94360 

•94456 

•94551 

19 

71 

•94551 

•94646 

•94739 

•94832 

•94924 

•95015 

•95105 

18 

72 

•95105 

•95195 

•95283 

•95371 

•95458 

•95545 

•95630 

17 

73 

•95630 

•95715 

•95798 

•95881 

•95964 

•96045 

•96126 

16 

74 

•96126 

•96205 

•96284 

•96363 

•96440 

•96516 

•96592 

15 

75 

•96592 

•96667 

•96741 

•96814 

•96887 

•96958 

•97029 

14 

76 

•97029 

•97099 

•97168 

•97236 

•97304 

•97371 

•97437 

13 

77 

•97437 

•97402 

•97566 

•97629 

•97692 

•97753 

•97814 

12 

78 

•97814 

•97874 

•97934 

•97992 

•98050 

•98106 

•98162 

11 

79 

•98162 

•98217 

•98272 

•98325 

•98378 

•98429 

•98480 

10 

80 

•98480 

•98530 

•98580 

•98628 

98676 

•98722 

•98768 

9 

81 

•98768 

•98813 

•98858 

•98901 

•98944 

•98985 

•99026 

8 

82 

•99026 

•99066 

•99106 

•99144 

•99182 

•99218 

•99254 

7 

S3 

•99254 

•99289 

•99323 

•99357 

•99389 

•99421 

•99452 

6 

84 

•99452 

•99482 

•99511 

•99539 

•99567 

•99593 

•99619! 

5 

85 

•99619 

•99644 

•99668 

•99691 

•99714 

•99735 

•99756 

4 

86 

•99756 

•99776 

•99795 

•99813 

•99830 

•99847 

•99862 

3 

87 

•99862 

•99877 

•99891 

•99904 

•99917 

•99928 

•99939! 

2 

88 

•99939 

•99948 

•99957 

•99965 

•99972 

•99979 

•999841 

1 

89 

•99984 

•99989 

•99993 

•99996 

•99998 

•99999 

1-0000 

0 

_ -:J£ 

60' 

50' 

40' J 

30' 


10' 

0' 1 

Deg. 


Natural Cosine. 







































m 


Trigonometry. 


Natural Tangent* 


Deg. 

0' 

10' 

20' 

30' 

40' 

50' 

60' 


0 

•ooooo 

•00290 

•00581 

*00872 

*01163 

•01454 

•01745 

1 89 

1 

■01745 

•02036 

•02327 

•02618 

•02909 

•03200 

•03492 

88 

2 

•03492 

•03783 

•04074 

•04366 

•04657 

•04949 

•65210 

! 87 

3 

•05240 

•05532 

•05824 

•06116 

•06408 

•06700 

•06992 

{ 86 

4 

•06992 

•07285 

•07577 

•07870 

•08162 

•08455 

•08748 

1; 85 

5 

•08748 

•09042 

•09335 

•09628 

•09922 

•10216 

•10510 

! S4 

6 

*10510 

•10804 

•11098 

•11393 

•11688 

•11983 

•12278 

! 83 

7 

*■•12278 

•12573 

•12869 

•13165 

•13461 

I -13757 

•14054 

1 82 

8 

•14054 

•14350 

•14647 

•14945 

•15242 

•15540 

•15838 

81 

9 

•15838 

•16136 

•16435 

•16734 

*1 7 033 

•17332 

•17632 

80 

10 

•17632 

•17932 

•18233 

•18533 

•18834 

•19136 

•19438 

79 

11 

•1943S 

•19740 

•20042 

•20345 

•20648 

•20951 

•21255 

78 

12 

•21255 

•21559 

•21S64 

•22169 

•22474 

•22780 

•23086 

77 

13 

•23086 

•23393 

•23700 

•24207 

•24315 

•24624 

•24932 

76 

14 

•24932 

•25242 

•25551 

•25861 

•26172 

•26483 

•26794 

75 

15 

•26794 

•27106 

•27419 

•27732 

•28045 

•28359 

•28674 

74 

16 

•28674 

•28989 

•29305 

•29621 

•29938 

•30255 

•30573 

73 

17 

•30573 

•30891 

•31210 

•31529 

•31849 

•32170 

•32491 

72 

18 

•32491 

•32813 

•33136 

•33459 

•33783 

•34107 

•34432 

71 

19 

•34432 

•34758 

•35084 

•35411 

•35739 

•36067 

•36397 

70 

20 

•36397 

•36726 

•37057 

•37388 

•37720 

•38053 

•38386 

69 

21 

•38386 

•38720 

•39055 

•39391 

•39727 

•40064 

•40402 

68 

22 

•40402 

•40741 

•41080 

•41421 

•41762 

•42104 

•42447 

67 

23 

•42447 

•42791 

•43135 

•43481 

•43827 

•44174 

•44522 

66 

24 

•44522 

•44871 

•45221 

•45572 

•45924 

•46277 

•46630 

65 

25 

•46630 

•46985 

•47340 

•47697 

•48055 

•48413 

•48773 

64 

26 

•48773 

•49133 

•49495 

•49858 

•50221 

•50586 

•50952 

63 

27 

•50952 

•51319 

•51687 

•52056 

•52426 

•52798 

•53170 

62 

28 

•'53170 

•53544 

•53919 

•54295 

•54672 

•55051 

•55430 

61 

29 

•55430 

•55811 

•56193 

•56577 

•56961 

•57347 

•57735 

60 

30 

•57735 

•58123 

•58513 

•58904 

•59296 

•59690 

•60086 

59 

31 

•60086 

•60482 

•60880 

•61280 

•61680 

•62083 

•62486 

58 

32 

•62486 

•62892 

•63298 

•63707 

•64116 

•64527 

•64940 

57 

33 

•64940 

•65355 

•65771 

•661S8 

•66607 

•67028 

•67450 

56 

34 

•67450 

•67874 

•68300 

•68728 

•69157 

•69588 

•700201 

55 

35 

•70020 

•70455 

•70891 

•71329 

•71769 

•72210 

•726541 

54 

36 

•72654 

•73099 

•73546 

•73996 

•74447 

•74900 

530t) | 

53 

37 

•75355 

•75812 

•76271 

•76732 

•77195 

•77661 

•7812Sj 

52 

38 

•78128 

•78598 

•79069 

•79543 

•80019 

•80497 

•80978 

51 

39 

•S0978 

•S1461 

•81946 

•82433 

•82923 

•83415 

•83909 

50 

40 

•83909 

•84406 

•S4906 

•85108 

•85912 

•86414 

•86928 

49 

41 

•86928 

•S7440 

•87955 

•88172 

•88992 

•89515 

•90040 

48 

42 

•90040 

•90568 

•91099 

•91633 

•92169 

•92704 

93251 

47 

43 

•93251 

•93796 

•94345 

•94S96 

•95450 

•96008 

•9656S 

46 

44 

•96568 

•97532 

•97699 

•9S269 

•98843 

•99419 

1*0000: 

45 

, 

60' 

50' | 

L 

40' 

30' 

i 

20' 

10' 

j 

0' 

i 

Deg. 

r 



Natural Colati 

gent. 





L_. 



































Trigonometry. 


125 


Natural Tangent* 


Deg. 

0' 

10' 

20' 

30' 

40' 

50' 

60' 


45 

1-0000 

1-0058 

1-0117 

1-0176 

1-0235 

1-0295 

1-0355 

44 

46 

1-0355 

1-0415 

1-0476 

1-0537 

1-0599 

1-0661 

1-0723 

43 

47 

1-0723 

1-0786 

1-0849 

1-0913 

1-0977 

1-1041 

1-1106 

42 

48 

1-1100 

1-1171 

1-1236 

1-1302 

1-1369 

1-1436 

1-1503 

41 

49 

1-1503 

1-1571 

1-1639 

1-1708 

1-1777 

1-1847 

1-1917 

40 

50 

1-1917 

1-1988 

1-2059 

1-2130 

1-2203 

1-2275 

1-2348 

39 

51 

1-2848 

1-2422 

1-2496 

1-2571 

1-2647 

1-2722 

1-2799 

38 

52 

1-2799 

1*2876 

1-2954 

1-3032 

1-3111 

1-3190 

1-3270 

37 

53 

! -3270 

1-3351 

1-3432 

1-3514 

1-3596 

1-3679 

1-3763 

36 

54 

1-3763 

1-3848 

1-3933 

1-4019 

1-4106 

1-4193 

1-428] 

35 

55 

1-4281 

1-4370 

1-4459 

1-4550 

1-4641 

1-4732 

1-4825 

34 

56 

1-4825 

1-4919 

1-5013 

1-5108 

1-5204 

1-5301 

1-5398 

33 

57 

1-5398 

1-5497 

1-5596 

1-5696 

1-5798 

1-5900 

1-6003 

32 

58 

1-6003 

1-6107 

1-6212 

1-6318 

1-6425 

1-6533 

1-6642 

31 

59 

1 6612 

1-6752 

1-6864 

1-6976 

1-7090 

1-7204 

1-7320 

30 

60 

1-7320 

1-7437 

1-7555 

1-7674 

1-7795 

1-7917 

1-8040 

29 

61 

1-8040 

1-8164 

1-8290 

1-8417 

1-8546 

1-8676 

1-8807 

28 

62 

1-8807 

1-8939 

1-9074 

1-9209 

1-9347 

1-9485 

1-9626 

27 

63 

1-9626 

19768 

1-9911 

2-0056 

2-0203 

2-0352 

2-0503 

26 

64 

2-0503 

2-0655 

2-0809 

2-0965 

2-1123 

2-1283 

2-1445 

25 

65 

2-1445 

2*1608 

2-1774 

2-1942 

2-2113 

2-2285 

2-2460 

24 

66 

2-2^60 

2-2637 • 

2-2816 

2-2998 

2-3182 

2-3369 

2-3558 

23 

67 

2-3558 

2-3750 

2-3944 

2-4142 

2-4342 

2-4545 

2-4750 

22 

68 

2-4750 

2-4959 

2-5171 

2-5386 

2-5604 

2-5826 

2-6050 

21 

69 

2-6050 

2-6279 

2-6510 

2-6746 

2-6985 

2-7228 

2*7474 

20 

70 

2-7474 

2-7725 

2-7980 

2-8239 

2-8502 

2-8769 

2-9042 

19 

71 

2-9042 

2-9318 

2-9600 

2-9886 

3-0178 

3-0474 

3-0776 

18 

72 

3-0776 

3*1084 

3-1397 

3-1715 

3-2040 

3-2371 

3-2708 

17 

73 

3-2708 

3-3052 

3-3402 

3-3759 

3-4123 

3-4495 

3-4874 

16 

74 

3-4874 

3-5260 

3-5655 

3-6058 

3-6470 

3-6890 

3-7320 

15 

75 

3-7320 

3-7759 

3-8208 

3-8667 

3-9136 

3-9616 

4-0107 

14 

76 

4-0107 

4-0610 

4-1125 

4-1652 

4-2193 

4-2747 

4-3314 

13 

77 

4-3314 

4-3896 

4-4494 

4-5107 

4-5736 

4-6382 

4-7046 

12 

78 

4-7046 

4-7728 

4-8430 

4-9151 

4-9894 

5-0658 

5-1445 

11 

79 

5-1445 

5-2256 

5-3092 

5-3955 

5-4845 

5-5763 

5-6712 

10 

SO 

5-6712 

5-7693 

5-8708 

5-9757 

6-0844 

6-1970 

6-3137 

9 

81 

6-3137 

6-4348 

6-5605 

6-6011 

6-8269 

6-9682' 

7-1153 

8 

82 

7-1153 

7-2687 

7-4287 

7-5957 

7-7703 

7-9530 

8-1443 

7 

83 

8-1443 

8-3449 

8-5555 

8-7768 

9-0098 

9-2553 

9-5143 

6 

84 

9-5143 

9-7881 

10-078 

10-385 

10-711 

11-059 

11-430 

5 

85 

11-430 

11-826 

12-250 

12-760 

13-196 

13-726 

14-300 

4 

86 

14-300 

14-924 

15-604 

16-349 

17-169 

18-074 

19-081 

3 

87 

19-081 

20-205 

21-470 

22-003 

24-541 

26-431 

28-636 

2 

88 

28-636 

31-241 

34-367 

38-188 

42-964 

49-103 

57-289 

1 

89 

57-289 

68-750 

85-939 

114-58 

171-88 

343-77 

oo 

0 


60' 

50' 

_J 

40' 

30' 

20' 

10' 

.0' 

Deg. 


Natural Cotangent. 


li * 






























Triuonometrt. 


126 





Natural Secant* 




Deg. 

0' 

10' 

20' 

30' 

40' 

50' 

60' 

1 

0 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0001 

1.0001 

89 

1 

1.0001 

1.0002 

1.0002 

1.0003 

1.0004 

1.0005 

1.0006 

88 

2 

1.0006 

1.0007 

1.0008 

1.0009 

1.0010 

1.0012 

1.0013 

87 

3 

1.0013 

1.0015 

1.0016 

1.0018 

1.0020 

1.0022 

1.0021 

86 

4 

1.0024 

1.0026 

1.0028 

1.0031 

1.0033 

1.0035 

1.003S 

85 

5 

1.0038 

1.0040 

1.0043 

1.0046 

1.0049 

1.0052 

1.0055 

84 

6 

1.0055 

1.0058 

1.0061 

1.0064 

1.0068 

1.0071 

1.0075 

83 

7 

1.0075 

1.0078 

1.0082 

1.00S6 

1.0090 

1.0094 

1.0098 

82 

8 

1.0098 

1.0102 

1.0106 

1.0111 

1.0115 

1.0120 

1.0124 

81 

9 

1.0124 

1.0129 

1.0134 

1.0139 

1.0144 

1.0149 

1.0154 

80 

10 

1.0154 

1.0159 

1.0164 

1.0170 

1.0175 

1.0181 

1.0187 

79 

11 

1.0187 

1.0192 

1.0198 

1.0204 

1.0210 

1.0217 

1.0223 

78 

12 

1.0223 

1.0229 

1.0236 

1.0242 

1.0249 

1.0256 

1.0263 

77 

13 

1.0263 

1.0269 

1.0277 

1 0284 

1.0291 

1.0298 

1.0306 

76 

14 

1.0306 

1.0313 

1.0321 

1.0329 

1.0336 

1.0344 

1.0352 

75 

15 

1.0352 

1.0360 

1.0369 

1.0377 

1.0385 

1.0394 

1.0403 

74 

16 

1.0403 

1.0411 

1.0420 

1.0429 

1.0438 

1.0447 

1.0456 

73 

17 

1.0456 

1.0466 

1.0475 

1.0485 

1.0494 

1.0504 

1.0514 

72 

18 

1.0514 

1.0524 

1.0534 

1.0544 

1.0555 

1.0565 

1.0576 

71 

19 

1.0576 

1.0586 

1.0597 

1.0608 

1.0619 

1.0630 

1.0641 

70 

20 

1.0641 

1.0653 

1.0664 

1.0676 

1.0687 

1.0699 

1.0711 

69 

21 

1.0711 

1.0723 

1.0735 

1.0747 

1.0760 

1.0772 

1.Q7S5 

68 

22 

1.0785 

1.0798 

1.0810 

1.0823 

1.0837 

1.0850 

1.0863 

67 

23 

1.0863 

1.0877 

1.0890 

1.0904 

1.0918 

1.0932 

1.0946 

66 

24 

1.0946 

1.0960 

1.0974 

1.0989 

1.1004 

1.1018 

1.1033 

65 

25 

1.1033 

1.1048 

1.1063 

1.1079 

1.1094 

1.1110 

1.1126 

64 

26 

1.1126 

1.1141 

1.1157 

1.1174 

1.1190 

1.1206 

1.1223 

63 

27 

1.1223 

1.1239 

1.1256 

1.1273 

1.1290 

1.1308 

1.1325 

62 

28 

1.1325 

1.1343 

1.1361 

1.1378 

1.1396 

1.1415 

1.1433 

61 

29 

1.1433 

1.1452 

1.1470 

1.1489 

1.1508 

1.1527 

1.1547 

60 

30 

1.1547 

1.1566 

1.1586 

1.1605 

1.1625 

1.1646 

1.1666 

59 

31 

1.1666 

1.1686 

1.1707 

1.1728 

1.1749 

1.1770 

1.1791 

58 

32 

1.1791 

1.1833 

1.1 S35 

1.1856 

1.1878 

1.1901 

1.1923 

57 

33 

1.1923 

1.1946 

1.1969 

1.1992 

1.2015 

1.2038 

1.2062 

56 i 

34 

1.2062 

1.2085 

1.2109 

1.2134 

1.2158 

1.2182 

1.2207 

55 

35 

1.2207 

1.2232 

1.2257 

1.2283 

1.2308 

1.2334 

1.2360 

54 

36 

1.2360 

1.2386 

1.2413 

1.2440 

1.2466 

1.2494 

1.2521 

53 

37 

1.2521 

1.2548 

1.2576 

1.2604 

1.2632 

1.2661 

1.2690 

52 

38 

1.2690 

1.2719 

1.2748 

1.2777 

1.2807 

1.2S37 

1.2867 

51 1 

39 

1.2867 

1.2898 

1.2928 

1.2959 

1.2990 

1.3022 

1.3054 

50 i 

40 

1.3054 

1.3086 

1.3118 

1.3150 

1.3183 

1.3216 

1.3250 

49 

41 

1.3250 

1.3283 

1.3317 

1.3351 

1.3386 

1.3421 

1.3456 

48 

42 

1.3456 

1.3491 

1.3527 

1.3563 

1.3599 

1.3636 

1.3673 

47 

43 

1.3673 

1.3710 

1.3748 

1.3785 

1.3824 

1.3862 

1.3901 

46 

44 

1.3901 

1.3940 

1.39S0 

1.4020 

1.4060 

1.4101 

1.4142 

45 


60/ 

_ 

50' 

40' 

30' 

20' 

10' 

0' 

Deg. 

Natural Cosecant. 



























Trigonometry. 


127 


r 


Natural Secant* 


Deg. 

0- 

10' 

20' 

30' 

40' 

50' 

60' 


45 

1-4142 

1-4183 

1-4225 

1-4267 

1-4309 

1-4352 

1-4395 

44 

46 

1-4395 

1-4439 

1-4483 

1-4527 

1-4572 

1-4617 

1-4662 

43 

47 

1-4662 

1-4708 

1-4755 

1-4801 

1-4849 

1*4896 

1-4944 

42 

48 

1-4944 

1-4993 

1-5042 

1-5091 

1-5141 

1-5191 

1-5242 

41 

49 

1-5242 

1-5293 

1-5345 

1-5397 

1*5450 

1-5503 

1-5557 

40 

50 

1-5557 

1-5611 

1-5666 

1-5721 

1-5777 

1-5833 

1-5890 

39 

51 

1-5890 

1-5947 

1-6005 

1-6063 

1-6122 

1*6182 

1-6242 

38 

52 

1-6242 

1-6303 

1*6364 

1-6426 

1-6489 

1-6552 

1-6616 

37 

53 

1-6616 

1-6680 

1-6745 

1-6811 

1-6878 

1-6945 

1-7013 

36 

54 

1-7013 

1-7081 

1-7150 

1-7220 

1-7291 

1-7362 

1-7434 

35 

55 

1-7434 

1-7507 

1-7580 

1-7655 

1-7730 

1-7806 

1-7882 

34 

56 

1-7882 

1-7960 

1-8038 

1-8118 

1-8198 

1-8278 

1-8360 

33 

57 

1-8360 

1-8443 

1-8527 

1-8611 

1-8697 

1-8783 

1-8870 

32 

58 

1-8870 

1-8959 

1-9048 

1-9138 

1-9230 

1-9322 

1-9416 

31 

59 

1-9416 

1-9510 

1-9606 

1-9702 

1-9800 

1-9899 

2-0000 

30 

60 

2-0000 

2-0101 

2-0203 

2-0307 

2-0412 

2-0519 

2-0626 

29 

61 

2-0626 

2-0735 

2-0845 

2-0957 

2-1070 

2-1184 

2-1300 

28 

62 

2-1300 

2-1417 

2-1536 

2-1656 

2-1778 

2-1901 

2-2026 

27 

63 

2-2026 

2-2153 

2-2281 

2-2411 

2-2543 

2-2676 

2-2811 

26 

64 

2 2811 

2-2948 

2-3087 

2-3228 

2-3370 

2-3515 

2-3662 

25 

65 

2.3662 

2-3810 

2-3961 

2-4114 

2-4269 

2-4426 

2-4585 

24 

66 

2-4585 

2-4747 

2-4911 

2-5078 

2-5247 

2-5418 

2-5593 

23 

67 

2-5593 

2-5769 

25949 

2-6131 

2-6316 

2-6503 

2-6694 

22 

68 

2-6691 

2-6888 

2-7085 

2-7285 

2-7488 

2-7694 

2-7904 

21 

69 

2-7904 

2-8117 

2-8334 

2-8554 

2-8778 

2-9006 

2-9238 

20 

70 

2-9238 

2-9473 

2-9713 

2-9957 

3-0205 

3-0458 

3-0715 

19 

71 

3-0715 

3-0977 

3-1243 

3-1515 

3-1791 

3 2073 

3-2360 

IS 

72 

3-2360 

3-2653 

3-2951 

3-3255 

3-3564 

3-3880 

3-4203 

17 

73 

3-4203 

3-4531 

3-4867 

3-5209 

3-5558 

3-5915 

3-6279 

16 

74 

3-6279 

3-6651 

3-7031 

3-7419 

3-7816 

3-8222 

3-8637 

15 

75 

3-8637 

3-9061 

3-9495 

3-9939 

4-0393 

4-0859 

4-1335 

14 

76 

4-1335 

4-1823 

4-2323 

4-2836 

4-3362 

4-3900 

4-4454 

13 

77 

4-4454 

4-5021 

4-5604 

4-6202 

4-6816 

4-7448 

4-8097 

12 

78 

4-8097 

4-8764 

4-9451 

5-0158 

5.0886 

5-1635 

5-2408 

11 

79 

5-2408 

5-3204 

5-4026 

5-4874 

5.5749 

5-6653 

5-7587 

10 

SO 

5-7587 

5-8553 

5-9553 

6-0588 

6-1660 

6-2771 

6-3924 

9 

81 

6-3924 

6-5120 

6-6363 

6-7654 

6-8997 

7-0396 

7-1852 

8 

82 

7-1852 

7-3371 

7-4957 

7-6612 

7-8344 

8-0156 

8-2055 

7 

83 

8-2055 

8-4046 

8-6137 

8-8336 

9-0651 

9-3091 

9-5667 

6 

84 

9-5667 

9-8391 

10-127 

10.437 

10-758 

11-104 

11-473 

5 

S5 

11-473 

11-868 

12-291 

12.745 

13-234 

13-763 

14-335 

4 

86 

14-335 

14-957 

15-636 

16.380 

17-198 

18-102 

19-107 

3 

87 

19-107 

20-230 

21-493 

22.925 

24-562 

26-450 

28-653 

2 

88 

28-653 

31-257 

34-382 

38.201 

42-975 

49-114 

57*298 

1 

89 

57-298 

68-757 

85-945 

114.59 

171.88 

343-77 

CO 

0 


60' 

50' 

40' 

30' 

20' 

10' 

0' 

Deg. 

_ 


J 

1 




ll 



Natural Cosecant* 










































128 


Right-Angled Triangle. 


FORMULA FOR RIGHT-ANGLED TRIANGLES. 



b 


a = 

y/fr+C*, 

1, 

Q 

cg sin.2C 

4 

io, 

a = 

C 

sin. C’ 

2, 

Q 

= | 5’tan.C, 

ii, 


b 


Q 

= £ c* cot. C, 

12, 

a = 

cos. O* 

3, 

Q = hc 

:V\ a+c)(a — c) 

13, 

a — 

2 \ / • 

4, 

sin C 

c 

14, 


\/ sm.2C 



a ’ 


b = 

a cos. C, 

5,. 

cos .C 

b 

15, 





a 


b = 

c cot. C, 

6, 


c 





tan.C 

~ ~~T r 

16, 

b = 

a s'm.B, 

7 


b 


6 = 

c tan. B, 

8, 

«in.2C 

4 Q 

~ a 

17, 

b = 

rw 

9, 

tan.C 

_2Q 

* 

18, 


\/ tan.C’ 






Say the angle to he <7= 60°. In the first column of the table of sines, 60° 
corresponds with 0-86602 in the next column, which is the length of sin. 60°, 
when the radius of the circle is one, or the unit, and the expression sin. 60°X36 
means 0-86602X36 = 31-17672, and likewise with all the other Trigonometrical 
expressions. 

In a triangle the functions for an angle have a certain relation to the oppo¬ 
site side; it is this relationship which enables us to solve the triangle by the ap¬ 
plication of Simple Arithmetic. * 

In triangles the sides are denoted by the letters a, b, and c; their respective 
opposite angles are denoted by A, B, and G, and the area by Q. 


Example 1. Fig. 136 The side c in a right angled Triangle being 365 feet, and 
the angle C = 39° 20'. How long is the side a —? 

e 365 365 _ 

mar = = ontm = 575-86 feet > the answer - 


Formula 2. a 
















Obt . v ^ uf .- angled Triangle. 
















































/ 130 


Plane Trigonometrt. 


Example 2. Fig. 136. An inclined plane a 150 feet long, and c = 27 feet, the 
height over its base. What is the angle of inclination C = ? 


Formula 14, 


c 27 

sin.C = — = —- «= 0’18000. 
a 150 


Find 0-18000 in the table of sines, which will be found at 10° 30' which is the 
angle C nearly. 

Example 3, Fig. 137 An oblique angled triangle has the sides c = 27’6 feet, the 
angle C = 34° 10', and the angle A = 47° 40'. How long is the side a — ? 

_ 7 , c sin.A 27‘6Xsw»-47 p 40 „„ Q o * ± +t . 

Formula 1. a= — : —— = — . — = 36-33 feet, the answer. 


sin.C 


sin,34P 10' 


c 

A 


+ 


C — 


By Logarithms. 

log.a — log.c-\-log.sin.A — log.sin.C, 
log.27-6 = 1-44090 
log.sin.47°40' = 1:86878 
1-30968 

Iog.sin.34° 10' = 1:74942 

log. 36-4 = 1-56026, or a — 36*4 feet. 


Example. 1. Two ships of war notice a strong firing from a castle; in order to 
be safe, they keep themselves at a distance beyond the reach of the balls from 
the castle. To measure the distance from the castle, they place the vessels 800 
yards from each other, and observe the angles between the castle and the ves¬ 
sels to be A — 63° 45', B = 75° 50'. What will be the two distances from the 
castle ? 

C = 180 — 63° 45' — 75° 50' = 40° 25'. 

To A the distance will be, 


b = 


c sin.B _ 800XsiP- 75° 50 ' 
sin.C sin. 40° 25' 


= 1195-75 yards. 


To B the distance will be, 

c sin. A 
sin.C 


a — 


800Xsin. 63° 45 
sin. 40° 25' 


= 1106-6 yards. 


Example 2. From a window in the lower floor of a house which lays level with 
the foot of a tower, is observed an angle = 40° to the top of the tower. From 
another window in the upper story, in the same perpendicular as the lower 
window, the altitude of the tower is observed to be = 37° 30', which is 18 feet 
above the lower window. 

Then we have A = 90 — 40° = 50°. C = 90+37° 30' = 127° 30'. 

B = 180 — 50 — 127° 39' = 2° 30'. 6 = 18 feet. 

What will be the height of the tower and the distance from it? 

The distance from the lower window to the top of the tower = c. 


c = 


b sin.C 18Xsjn.l27° 35' 


sin.B 


sin.2° 30' 


= 327-3 feet. 


The height of the tower = h. 

h = c. sin.A = 327"3Xsin.40° = 210 feet. 
The distance to the tower = d. 

d = c. eos.A = 327 - 3Xcos,40° = 250-8 feet. 



















r late! 11 




































































































































































































































































































































































































































































































































































































































































4 } 




















> ' ’ 

. • ■ 




' 




















- • 






<v 














■ - w . 




; •- • m; ./<■ 


- - ■ ■ ■' 










Spherical Trigonometry. 


in 


To Solve Triangles Medianically. 

PLATE III. 

The accompanying diagram is so constructed that the moveable arm repre¬ 
sents the hypothenuse, the square lines the two sides, and the circular scale the 
angles in a right-angled triangle. The scale numbered from the centre towards 
0 will be called b, and the one at right angle to be called c. 

Example 1. Let the lines that form the right-angle be given as b = 12 and 
c = 4 inches. Required the hypothenuse a, and the angles B and Ct 

Find where the two lines 12 and 4 crosses each other, move the arm to this 
crossing-point, which then indicates the length of the hypothenuse = 12'65 on 
the arm a ; the two angles will be found at a on the scale. B — 71° 40' and 
C = 18° 2<y. 

If one angle and a side is given, set the arm on the given angle, and the in¬ 
tersection of the given side with the arm shows the length of the hypothenuse 
and the other side. 

An oblique angled triangle can be two right angled triangles by drawing a 
: line from the largest angle perpendicular to the opposite side, and can be solved 
i by this diagram. 

Example 2. An oblique angled triangle being a — 65 feet, C = 34° 30' and 
B = 68° 20'. 

Required tiie two sides b and c ? 

Set the arm on the given angle 34° 30', and at 65 feet on the arm will be 
found the height of the triangle = 37 feet on scale c, and one part of the side b 
is 54 feet on the scale b. 

From the given angle B = 68° 20' 

Subtract the complement of 34° 30' = 55° 30' 

Set the arm on the angle, 12° 50' 

Now at the height 37 on the scale b will be found 38 feet on the arm, which is 
the length of the side c, and the other part of the side b is 9 feet on the scale c, 
then b = 54+9 = 63 feet. 

In a similar manner any plane triangle can be so solved. By a little practice, 
this Table is very useful for approximating triangles. 


>«- 


SPHERICAL TRIGONOMETRY. 

Splierfcal Trigonometry treats of triangles which are drawn (or ima- 
gined to be) on the surface of a sphere; their sides are arcs of the great circle 
of the sphere, and measures by the angle of the arc. Therefore the trigonome¬ 
trical functions bear quite a different relation to the sides. 

Every section of a sphere cut by a plane is a circle. A line drawn through the 
centre and at right angles to the sectional circle is called an axis, and the two 
points where the axis meets the surface of the sphere are called the poles of the 
sectional circle. 

JST 139 

When the cutting plane goes through the 
centre of the sphere, it will pass through the 
great circle, and is then called the Equa¬ 
tor for the poles. Axis = N.S. Equator — 
G.E.T.W. 

Three great circle-planes, aa'a"a'", bb'b", 
and cc'c". cutting a sphere, NESW, will form 
a solid angle at the centre O, and a triangle 
ABC on the surface of the sphere, in which 
the arcs a, b, c, are the siles. The angles form¬ 
ed by each two planes are congruent to each 
of the appertinent angles A, B, and C. 





















132 


Right-Angled Spherical Triangle 




The sum of the three angles in a spherical triangle is greater than two right 
angles, and less than six right angles. 

By Spherical Trigonometry we ascertain distances and courses on the surface 
of the earth; positions and motions of the heavenly bodies, &c,, &c. Examples 
will be furnished in Geography and Astronomy. 

Example 1. Fig. 140 In a right-angled spherical triangle the side or hypotlae- 
nuse a = 36° 20', the angle B = 68° 50'. How long is the side b — ? 

Formula 1. sin.6 = sin.a.sin. 7? = sin..3fi°20'Xsm,G8°60'. 
a log.sin. 36° 20' = 1:77267 

B log.sin. 68° 50' — 1:96966 

The answer, log.sin. 33° 32' = 1:74233 orb — 33° 32'. 


sin./) = 

sin.a sin._B, 

1, 

tan.c = 

tan.a cos.5, 

2, 

cot.C ^ 

cos.a tan.5, 

a. 

tan.c = 

sin./) tan.C, 

4, 

cos. a = 

cos./> cos.c, 

5, 

COS.jB = 

cos -.b sin.C, 

6, 

tan.a = 

tan./) 
tan. C’ 

7, 

sin.c = 

tan./) 

8, 

tan .B ’ 

sin. a = 

sin. b 
tan. B’ 

9, 

sin.C = 

cos. E 

10, 

cos .b ’ 

cos.c = 

cos. a 

cos ./>’ 

11, 


d sin./? 
sm.ij = --— , 
sm. a 


sin. C = 


tan.C = 


tan. b 
tan.a 7 

tan.c 

sin./)’ 


tan.i? = — 


tan. 6 


, 


sm.c 


cos.c = 


cos .C 
sin. B’ 


sin. 


cos. a = 


tan.B’ 

























Oblique-Angled Spherical Triangle. 


133 


OBLIQUE-ANGLED SPHERICAL TRIANGLE. 



sin.a : sin.i = sin.vl : sin. B, 
sin.6 : sin.c = sin.I? : sin.C, 


sin.fl = 


sin. b 


sin.6 sin .A 


sin.13 ’ 

sin.c sin .B 


sin.C 


, . x t cos .h{A — B) 
tan.(a+i) = tan.ic - 

tan. (a— b) = tan.c ■ p j, 

v ' sm.i(A + By 

tan .i(A+B) = cot .\A C0S :Mj > ~ c ) 

^ 1 COS.4(0-l-C) ’ 

tan.±(A-£) = cot-M ) 

v 1 sin.A(6+c) * 


Cot.iA = tan.£(i? —• C) 


sin4(<ji-fc) 
sin.i(/>— c) 


, ,. sin. 3 (A+B) 

tan.^c = tan.i(a— b) 


19, 

20 , 

- 21 , 
22 , 

- 23, 
24, 

- 25, 
✓ 26, 


Example 2. Fig. HI Oblique angled spherical triangle, c = 72° 30'. B = 17° 30'. 
C = 79° 50'. 

How long is the side b — 1 

7 . , sin.c sin.B sin.72° S0'Xsin.I7° 30' 

Formula 20. sin .b = =- 7i^W - 

c -J- log.sin. 72° 30' = 1:97942 

B + log.sin. 17° 30' = 1:47812 

4- = 11:45754 

C + log. sin. 79° 50' = 1:99312 

The answer log.sin. 16° 5G' = 1:40442 or &— 16° 56.' 


12 























134 


Oblique Angler Spheric at, Triangle 


OBLIQUE-ANGLED SPHERICAL TRIANGLE. 



tan. 5 (/w+«)tan£(m — n = tan.£(a c)tani(« 
tan .m, = tan.c cos. A, - - - 

tan C = s * D ‘ m tan. A 
sin.(6 — m) 

_ cos.c cos. (b —*m) 
cos .m ’ 


— c) 


cos.a 


„ cos.a cos .m 

COS >11 — -, 

cos.c 
5 = m+n. 
cos.c tan .A 


cot .m = 


tan. a 

s = a+6+c = A-rB+C y 

s - 


in.iA - . / sin. (.5 c) sin. (5 6) 

V sin.6 sin.c 

/"cos.S cos.(S — X) 

Sin.T;d = \ / ■ n ■ 7 ^ i 

\y sin.ilsm.C ’ 


27, 

- 28, 

29, 

- 30, 

31, 

- 32, 
33, 


To Find the Area of a Spherical Triangle* 

Let Q be the area of the triangle in square degrees; if R — radius of the 
sphere, the length of one degree will be, 

27lR . ffa 


= 800 ’ or0ne S(iuare degree = 3285-58 • 

, . ^ cot.ic cotAa+cos.B 
cot AQ = -j 5 -, 


sin.iQ 


sin. 2? 

sin.^c sin.ia sin .B 
cos -ib 


1, 

2 , 































CONIC SECTIONS. 

A Conic Sectioij is the section obtained when a plane cuts a cone. 

The conic sections are of five different kinds, namely. 

1st. Triangle. When the plane cuts the cone through its axis. 

2d. Circle. When the plane cuts the cone at right angles to its axis. 

3d. Ellipse. When the plane cuts the cone obliquely passing through the two 
sides. 

4th. Parabola. When the plane cuts the cone parallel to one side. 

5th. Hyperbola. When the plane cuts the cone at an angle to the axis less than 
the angle of the axis and the side of the cone. 

The position of a point in a plane surface is determined by its course and dis¬ 
tance from a given point on a straight line, or by its distances from two lines 
inclined to one another 

Let AB and CD be two infinite lines inclined to 
one another, and P being a point in the same plane 
as the lines. It is evident that the position of this 
point P is determined by the distances x and y from 
the lines AB and CD. Those lines and distances 
are called, AB the axis of ordinate, and CD the 
axis of abscissa, y the ordinate, x the abscissa, and 
o the origin. 

The abscissa x is commonly taken on the absciss’ axis. Now take different 
values of the abscissa x, and by some formula or rule calculate the ordinate y; 
then a number of points P,P'P", &c., may be obtained; join those points by a 
line, then the rule or formula is called an equation for that line. Equations 
of this kind will here be furnished for the curve in the conic sections. 

Transverse-axis is the longest line that can be drawn in an Ellipse. 

Conjugate-axis is a line drawn through the centre, at right angles to the 
transverse axis. 

Parameter of any diameter is a third proportional to that diameter, and its 
conjugate. 

Focus is the point in the axis where the ordinate is equal to half the di¬ 
ameter. 




> J 


144. 


Cycloid. 


y — 0-637 V oc(7td — oc) t 
e = l-211 d, p-0-md. 






















136 


Conic Sections. 



CONIC SECTIONS, 


a : b = sin.zu : sin.u, 

k 

a sin.u 

i 


sin.zo ’ 

j-j 

x : c — sin.zo : sin..?, 


x sin.? 

2, 

c — 

sin.to ’ 

x : d = sin.zo : sin.(?+u), 

d-~ 

x sin.(?+u) 
sin. w 

3, 


y,= 


a?.sin.z ,ci sin.(£+u) 
sin.to ' sin.to 


a sin.u 

4 -=- 

sin. to 


), 


5, 


„ a?.sin.z 

V = —•—S— 

J sin. w 

This is the general formula for all conic sections. 


(a sin.[z+u] + a sin.u). 


6 , 


In any conic section, a point Pcan be calculated by this formula 6, but for 
the different sections, it will he found greatly simplified on the next pages. 

For a Parabola z-{-v — 180, therefore sin .2 = sin.a, and 


x sin. 3 u 
sin.-zu * 





























Conic Sections. 137 



12 * 













































138 


Conic Sections. 





















































Mechanics.—Statics. 


139 


MECHANICS. 

Mechanics is that branch of Natural Philosophy which treats of the action 
of force, motion, and power. Mechanics is divided into four parts, namely, 

Statics the science of forces in equilibrium. 

Dynamics, the science of forces in motion, it produces power or effect. 

Hydrostatics, the science of fluids in equilibrium 

hydrodynamics the science of fluids in motion, its causes, power or effects. 

Statics* Lever* Momentum. 

Lever is an inflexible bar, supported in one point called the Fulcrum, oi 
centre of motion. The length of a lever is measured from the fulcrum to where 
the force or resistance acts, (when the force acts at right angles to the lever) or, 
the length of a lever is measured from the fulcrum at right angles to the direc¬ 
tion of the force. 

W— Weight, and l — lever for IP) „ 

F = Force, and L = lever for F \ See Fl S- 156 - 

Momentum is the product of force or weight, multiplied by the length of 
the lever it acts upon. 

The products Wl and FL are called Statics Momentums; when these mo- 
mentums are equal there will be no motion, and the weight IF will balance 
the force F. When one momentum is greater than the other, there will be a 
motion, and the velocity of that motion is measured by the difference of the 
momentums. 

Levers are of three distinct kinds, with reference to the relative positions of 
the Force F, Weight W, and Fulcrum C. 

1st. Fulcrum C, is between the force F, and the weight W. 

2d. Weight W, is between the fulcrum C, and the force F. 

3d. Force F, is between the f ulcrum C , and the weight W. 

Example 1. Figure 156. The weight W — 68 pounds, the lever l — 3*86 feet, 
andZ= 10 feet 6 inches. 

Required the force F = ? 


Formula 1. F = - 


Wl 68X3-86 


10'5 


= 25 pounds nearly. 


a = distance between the force F and the weight W. 

The formula 3, 4, 7, 8, 11, 12, are for finding the fulcrum C, when the force 
F, weight W, and the distance a, are given. 

Example 2. Fig. 157. The force F= 360 pounds, W — 1870, and a = 8 feet, 4 
inches. 

Required the position of the fulcrum c ? 

, „ Fr 360X8-333 2999-988 

Formula 7.1 = 


W—F 1870 —360 1510 

L = S-333+19'86 = 28-193 feet, the answer. 
Example 3. Fig. 161.The weight of the lever is Q = 18 pounds, 
gravity is x = 2-25 feet from the fulcrur 
L= 11-95. 

Required the force F = ? in pounds. 


= 19-86 feet. 


The centre of 
W = 299 pounds, l = 5 - 5 feet, and 


F= 


}V7 _ q x 299X5-5 — 18X2-25 


= 134*25 pounds. 


11-95 

Inclined. Plane* 

Example 4. Fig. 180. A load W= 3466 pounds, is to be drawn up an inclined 
plane, l = 638 feet long, and h = 86 feet high. 

What force is required to keep the load on the inclined plane ? 

hW 86X3466 

= 467-2 pounds. 


F =~r ~ 


638 
















140 


Lever-anj> Static Momentum. 

















































Lever and Static Momentum. 


141 




A force 0 act in" (alone) on the body B, can move 
it to a in a unit of time, another force P is able to 
move it to b in the same time; now if the two 
forces act at the same time, they will move the 
body to c. c is the resultant of a and b. 




163 . 


F = 


W 


r r 


RR ’ 


FRR 

W = - 7 -> 

r r 


n — number of revolutions of the wheels, 

n : n' — F : R, v : v' — rr ': RR', 

V — velocity of W, F = velocity of F. 


164 . 


W r r'F 


F- . W = FRR 

R R'R' r ’ rr'r" ’ 

n : n" = Fr" : R R , v : v' = rr'r" : 
RR'R". 

T Ft" &C. = radii of the pinions. 

R RR"&C. — radii of the wheels. 


Wr 

R ’ 

Wr 
R ~ R 

RF 

RF 

r ’ 

r ~w- 


Let P and § represent the magnitudes and direc¬ 
tions of two forces which act to move the body B. 
By completing the parallelogram, there will be ob¬ 
tained a diagonal force F. whose magnitude and di¬ 
rection is equal to the sum P and Q. F is called 
the resultant of P and Q. 


166 - ' ' . 

If three or more forces act in different directions 
to move a body B , find the resultant of any two of 
hem, and consider it as a single force. Between 
"his and the next force find a second resultant. 
thus: P d. P are magnitudes and directions of 
the forces. P+ Q = r, r+P = P= P-f Q+R, or F 
is the magnitude and direction of the three forces, 
P, Q, and P. 


165 . 





































142 


Pullets. 















































V -V-- 


FuNTCUT.AR anti CATENARIAN. 145 






















































144 


Inclined Plane. 


L- 

/\' 

h 

v 

m p= W* = Wsin.tJ, 

w-™ lx. 

A sm.y 

IF/ _ 

H == - = W cos.v. 

b 

/iF 

jllr^ 


181. 

F = W sim(y-iV), 

w F 

' sin.(y-|-i/)’ 

W = W cos.(v+v'). 



182 ’ F IFsin.u 

JO — jr 

COS. V 

w _ F cos.v' 
sin.y * 

W = W (cos.y+sin.y. tam/). 

F* 

l^r 

Jw% 

^ Xdr & 

l 

183. 

To solve an Inclined Plane by diagrams* 

F = magnitude and direction of the 
force, which is obtained by completing 
the parallelogram. 

By calculation see Formula, Fig. 180. 

•® 

\ \ 

1 \, 


184. 

W = weight of the body, and direc-- 
tion of the force of gravity ; to be drawn 
i at right-angles to the base b, and F par¬ 
allel to F. 

By calculation see Formula, Fig. 181. 

Jwf- 

\ll}J> J 


185. 

w = the force with which the body 
presses against the plane, to be drawn 
at right-angles to the plane l; then the 
parallelogram is completed. 

By calculation see Formula, Fig. 182. 





































Wedge and Screws 


145 



186 


Wedge. 


F = 


F a 


F = 


FI 


l a 

F = force acquired to drive the wedge. 




187. 

Let the line .F represent the magnitude and di¬ 
rection of a force acting to move the body B on the 
line CD-, then the line a represents a part of F 
which presses the body B against CD, and the line 
b represents the magnitude of the force which 
actually moves the body B. 

b = \f F — ft 3 , b — F COS V. 


188. 


V: W = h : b = sin.u : cos.u = tan.?\ 

F = if- = W tan.v. F' = F. 
b 


v: - 


Fb 


h tan.u 


-- = F cot.u. 



189. Force by a Screw. 

P — Pitch of the screw, 

r — radius on which the force F acts. 

F : W= P '.In r. 




WP 
2 nr 


W — 


F2rrr 



190. Force by Compound Screws. 

P = Pitch of the large screw, 
p = Pitch of the endless screw. 

R = radius of spur-wheel for the endless 

flPTPW 

F : W = 4^ R r : P p. 


F = 


WP r 


W = 


FP rt Fr 


4ti2 R r* ” P p 

On the spur-wheel is a cylinder by which 
the weight W is wound up, the formula will 
r ' _ radius of the cylinder, and 

F : W' = P r : 2 n R r. 

w p r w 

r ~2nRr rr 































































146 


Mechanics.—Statics. 


Example 4. Fig. 184. A Cylinder of cast iron, weighing TV = 5245 pounds, is to 
be rolled up an inclined plane; the angles v = 18° 20' and v' — 8° KF 
"What force is required to keep the cylinder on the plane? 

F= W. sin.(i>+tO == stn.524oX(18° 20'+8° 100 = 2340 pounds. 
Example 5. Fig. 185. An iron ball which weighs 398 pounds, is tied to an in¬ 
clined plane with a rope; the angle of the rope and the inclined plane is 
v' = 16° 40', and' v = 14° 30'. What force is acting on the rope ? 


r = = 1W 

cos.r' cos. 16° 40' 


Example 6. Fig. 170. What force F is required to raise a weight W— 8469 
pounds, by a double moveable pulley ? 

F = ~TV== ^X8469 = 2117’25 pounds. 

Example 7. Fig. 173. How much weight can a force F — 269 pounds lift by 
three compound moveable pulleys ? 

W = 2 U F =t 2*X269 = 2152 pounds, the answer. 


Screw. 


Example 8. Fig. 189. What force is required to lift a weight TV — 16785 pounds, 
by a screw, With a pitch P = 0-125 feet, the lever being r ±= 5 feet. 4 inches ? 


WP 
2*r V 


16785X0-125 


2X»14X5-333 
Including friction the force F will he 

F W{P+fdrt ) 


= 62-62 pounds, the answer. 


2 71 r 

Find the friction f on page 155. d diameter of the screw. 


Wedge. 


Example 9. Fig. 186. The head of the wedge a 
l - - 16£ inches; the resistance to be separated is R = 
the force F = ? (Friction omitted.) 


= 3 inches, and length 
■ 4846 pounds. Required 


F = 


4846X3 


16-5 

Including friction the force F will be, 

a 

l 


= 881 pounds. 


F=± 


. 4 ? +/(*+£)] 


146 


in which the friction / is to be found on page 155. 

Catenaria. 

Example 9. An iron chain 256 feet long, weighing 1560 pounds, is to be sus¬ 
pended between two points in the same horizontal line, but 196 feet apart. 

How deep will the chain hang under the line of suspension, and with what 
force will the chain act at the points of suspension ? 

Figure and Formula 178. we have given, 

TV = £X1560 = 780 pounds, l = £X256 = 128 feet, and a = |X196 = 98 feet. 

h = 0-6525/128* — 98* = 53-73 feet, the required depth under the horizontal 
line. 

2 V 53*7^ 

= 1-096, or v = 44° 44', and 2v = 89° 28'. 


cot.u 


98 


The required force will be, 

F = 


780X s in.44° 44' 


sin.89° 28' 


= 549 pounds. 


















Dynamics. 


147 


DYNAMICS 


Velocity is a space passed through in a unit of time. 

In machinery velocities are measured in feet per second ; for Steamboats and 
Railroads in miles per hour. 

Circular or angular velocity is the number of revolutions a revolving body 
makes per unit of time (minute.) 

Velocity «f jllcn, 

A foot-soldier travels about 28 inches per step. 

In common time 90 steps per minute = 35 feet per secopd = 2‘4 miles per hour. 
In quick time 110 “ = 4 3 =-3 “ 

Double quick time ,140 <c .= 5-5 =; 3'75 £< 

A soldier occupies in the rank, a front of 20 inches, and 13 inches deep with¬ 
out knapsack ; the interval between the ranks is 13 inches. 

Average weight of men, 150 pounds each.. 

Five men can stand in a space of 1 square yard. 

Example 1. A man walked 450 feet in 75 seconds. .With what velocity did he 
walk? 

s 450 

Formula 1. v — t- = ~ = 6 feet per second. 

t 10 

Example 2. A body moves 368 feet in t = 8 seconds. What velocity has it? 

Formula 1. v — 7 - = = 46 feet per second. 

t O 

Example 3. The radius of a wheel is 4.feet and 4 inches; it makes 131 revol 
utious per minute. What is the velocity of a point in the circumference? 


Formula 18. 


v 


2 - r n 2X3-14X4-33X131 
60 ’ ' 60 


595 feet per second. 


Power is the product of force and velocity: that is, a force multiplied 
by the velocity with which it moves, is the power of fhe force. Force without 
velocity is no power. , « 

Force and Power are two distinct quantities. Power .can not be increased or 
diminished by meehanical means; but Force can be increased and diminished 
ad libitum. 

When a, force is in motioji, and increased or diminished by mechanical means, 
the alteration will be at the. expense of the,veloeity, so that the power .will re¬ 
main the same. 


Horse Power. 

The unit for the measurement of mechanical Powers is assumed to be the 
power of a horse; or a force of 33000 pounds, moved through a space of one 
•foot in one minute, or 550 pounds moved 1 foot in 1 second. Another unit for 
measuring small powers is, onepgund moved fhroughu space of one foot in one 
second; this unit is called effect. 

One horse power is = 550 effects. 

One man’s power is = 50 effects. 

One horse power is = 11 men s power. 

Example 4. A man-draws up a bucket of water which weighs 52 pounds, from 
the bottom of a well 83 feet deep, which space the bucket passes in 43 seconds. 

With what effect is that man working ? 


1 Formula 5. 


i>= 52 > 5 8 - = 100 Effects. 

t 43 










<S 


148 


Power, Force and Velocity. 


Dynamical Formula* 


Velocity 

s 

i * - , 

t ’ 

- 1, 

Space 

5 *= V t, - 

- 2, 

Time 

S 

t = - 

V 

* 3, 

Power 

P = v F, - 

* 4, 

Power 

Fs 

p “ T ’' 

* 5, 

Force 

F- - 

V 

- 6, 

Force 

Pt 

F =—,- 

s 

- T, 

Velocity 

P 

V — — , 

F 

- 8, 

Space 

II 

*15 

1 

- 9, 

Time 

p’ 

10, 


Fv 

Horse Power H = > 

11, 

F s 

Horse Power H — ,-r, - 

Di)0 ^ 

12, 

Force 

„ 550 H 

F = -, 

V 

13, 

Force 

„ 550 Ht 

F = --, 

s 

14, 

Velocity 

550 H 
® — ^ , 

15, 

Space 

550 Ht 

s ~ jr » 

16, 

Time 

Fs 

1 mTr 

17, 

Velocity 

2 Tt r n 

V 60~’ ’ 

18, 

Effect 

F 2 tlt n 

F 60 ’ 

19, 


w _ F2 Ttr n _ Fr n 
550x60 5250 ’ 


Letters denote. 

= Pmver in Effects. 

— Horse-power, number of. 

= Velocity of the force F in feet per second. 
t = Time in seconds. 

s — the space in feet which the force F passes through in the time t. 

Circular Motion. 

r = radnts of the circle in feet. 

« = number of revolutions per minute. 
















Dynamics. 


149 


Power of Steam Engines* 

When the power of a steam-engine is to he calculated, we have the following 
quantities given: namely, 

a =■= area of the steam-cylinder piston in square inches. 
d = diameter of the steam-cylinder in inches. 

= stroke of piston in feet. 
jP= effectual pressure of steam per square inch. 
n — number of double strokes of piston per minute. 
f = a coefficient for frictions in the steam-engine. 

Example 5. The piston in a steam engine moves a space of 90 :feet In 25 sec* 
onds. with a force of 25000 pounds, {pressure of steam.) 

How many horse power is the engine ? 

Fs _ 25000X90 
5501! 


Formula 12. 


// = — 


163*6 horses. 


to 


550X^5 

The formula for the actual horse power of a steam-engine will appear, as, 

IT = 2 - a S P - ™ (1 —/.) .... - -2*1. 

ooOOO v J ' 

For condensing-engines in good order, the friction can be estimated 


For high pressure, 

By insertion of these values we have for 

ti S P n 

Condensing Engines H — - 

a SPn 


f = 0*32 
/= 0*25 


22 , 

High Pressure Engines H — . 23 > 

For condensing-engines, the vacuum must beinduded in the pressure P. 
Example 6. .Required the horse-power of a steam-engine, of the following di¬ 
mensions. 

a = 1017*8 square inches, area of piston. 

S = 4 feet stroke of piston. 

Pressured© the boiler 35 pounds per square in. * 

Vacuum 11 pounds. 

P = 85+11 = 46 pounds per square inch effectual pressure. 
n — 45 revolutions per minute. Then, 


Formula 22. 




1017*8X4X46X45 

24250 


= 347*5’horses. 


Example 7. Required the horse power of a high pressure-engine with the 
following dimensions ? 

a = 572*55 sq. in. s = 3 feet, p = 42 pounds, and n — 38 revolutions. 

572*55X3X42X58 
Formula 23. H = - 


19000- =220io CS ea. 

If the horse power is given, the other quantities will be ascertained by 

Hk 
SPn' 


a = 


24, 


S: 


Hk 


a HrC 


- 25, 


Hk 
a S n’ 

26, 

Hk 

n — „ , • 

a & »’ 

- 27. 


k = the coefficient for high or low pressure, formula 22 and 23. 

Nominal Horse Power* 

H = - 2 ^ . Established by Watt. 

47 




















150 


Observed Results of Power. 


OBSERVED RESULTS OF POWER. 



Work 

Force. 

Velocity. 

Effects 

Horses. 

Description of works. 

hours 
pa- day. 

F 

V 

P 

H 

A man can raise a weight by a single 



0'S 

40 

0-072 

fixed pulley, 

6 

50 

A man working a crank, 

A man on a tread-wheel, (horizontal), 

8 

20 

2'5 

50 

0-090 

8 

144 

0-5 

72 

0-130 

A man in a tread-wheel, (axis 24° from 




69 

0-125 

vertical), 

8 

30 

2-3 

A man draws or pushes in a horizontal 




60 

0-109 

direction. 

8 

30 

2 

A man pulls up or down, 

8 

12 

3-7 

44-4 

o-oso 

A man can bear on his back, 

A horse in a horse-mill, walking mode- 

7 

95 

2-5 

237 -5 





318 

0-577 

rately, 

8 

106 

3 

do. do. do. running fast, 

5 

72 

9 

648 

0-165 

An ox do. do, walking mod- 




308 

0-558 

erately, 

8 

154 

2 

A mule do. do. do. 

8 

71 

3 

213 

0-3S0 

An ass do. do. do. 

8 

33 

2-65 

87-4 

0-160 

Flour Mills* 






For every 100 pounds of fine flour ground per hour, requires 

550 

1-000 

One pair of mill-stones of 4 feet diameter making 120 rev. per 
minute, can grind 5 bushels of wheat to fine flour per hour, 

2400 

4-36 

Do. rye to coarse flour, 

Saw Mills, alternative* 

For every 20 square feet sawed per hour, in dry oak, there re¬ 
quires, 

Dry pine 30 square feet per hour, 

Circular Saw* 

A saw 2 - 5 feet in diameter; and making 270 revolutions per 
minute will saw 40 square feet in oak per hour, with 

1600 

550 

550 

550 

550 

2*91 

1-000 

1*000 

1-000 

1-000 

In dry spruce, 10 square feet per nour. 




Threshing Machine, 

Velocity of the feed rollers at the circumference 0*55 feet per sec¬ 
ond, Diameter of threshing-cylinder 3-5 feet and feet long, 
making 300 revolutions per minute, can thresh from 30 to 4C 
bushels of oats, and from 25 to 35 bushels of wheat, per hour. 

One man by a flail can thrash half a bushel per hour, (wheat.) 

2200 

4-000 

70 

0-127 


Paildle“\Vlii'cl Steamers. Observed Results. 


Length 


Draft 


Dia, d: length 

Miles 

Hors 

in 

Beam,. 

of 

Tonnage. 

of 

per 

Power. 

load line. 


water. 


Paddles. 

hour. 


100 

17 

4-5 

110 

16X4-25 

9 

115 

114 

22-5 

7.75 

241 

18.5X6-33 

7-5 

128 

128 

24 

7 

250 

15-75X5-5 

11-5 

130 

160 

27 

8-25 

570 

20X7-5 

10 

250 

200 

33 

13 

1100 

29X9-25 

. 14 

573 


Propeller Steamers* Observed Results* 


Length 


Draft 

Tonnage 

Dia. <£• Pitch 

Miles 

Horse 

in 

Beam. 

of 

of 

of 

per 

Power 

load line. 


water. 

displacement. 

Propeller. 

hour. 


112 

22 

»• 

t 

208 

8X18 

7 

76 

230 

32 

16 

1770 

14X32 

12 

650 

210 

38 

16-66 

2150 

14-5X36 

11 

500 



































Observed Results of Power.—Collision of Bodies in Motion. 


151 


DREDGING MACHINERY. 

Letters denote. 

T = tons of materials excavated per hour. 

h = height in feet, to which the materials are raised above the bottom of the 
excavated channel. 

/c = coefficient of the different materials. 

For very hard clay mixed with gravel, * - - k = Off 

“ hard pure clay, ------ - k = (H)7 

“ common clay or sand, - - - k — 0*05 

“ soft clay or loose sand - - - - - - k = 0 - 04 

“ very soft and loose do., - - - - - - k = 0*0? 

H — horse power required for the excavation. 

Example. What power is required to excavate 160 tons of hard pure clay per 
hour, and raise it up 25 feet ? 

For hard pure clay k — 0’07, then, 

25 

H— lC0^^j+0-07 J = 16*9, or 17 horses, nearly. 

The velocity of the buckets should be about 1 foot per second. 


COLLISION OF BODIES IN MOTION. 

When bodies in motion come in collision with each other, the sum of this 
concentrated momentum will be the same after the collision as before, but 
their velocities and sometimes their directions will differ. 

On the accompanying page the bodies are supposed to move in the same 
straight line, and the formula illustrates the consequences after collision. 

Letters denote. 

M and m — weight of the bodies in pounds. 

Fand v = their respective velocities in feet per second. 

V' and v' — respective velocities of the bodies after impact. 

When the bodies are perfectly hard or non-elastic their velocities after 
impact will be common == v r . When the bodies are partly elastic the letters 
k and k' = coefficients for their elasticity. 


For the body M } 


For the body m, 


k 


V = 


MV 


M(V— V') 
m v 

to ( i / — vy 


Example 1. Fig.193.The non-elastic body weighs M = 25 pounds, and moves 
at a velocity V— 12 feet per second; to = 16 pounds, and v = 9. Required the 
bodies’ common velocities, v' = ? after impact. 


v' = 


M F+ mv 25X12+16X9 


J/+m 


25+16 


= 10‘83 feet per second. 



















152 


Impact of Bodies. Dynamics. 



The bodies perfectly elastic. :j The bodies perfectly hard. 

























Impact op Bodies, Dynamics. 


If 3 



/ ) 
tA' > 



o © 





M 


/ 

/ 

I l y 

'+ .yt^o 


Y/bn) 


199. 

The bodies move in the same direction. 


F 


— 


F(ikf— kni)+vm{\+k) 

M+m * 

MV(l+k')+v(m — k'M) 


M+ 


rn 


200 . 

The bodies move in opposite directions, 
V[M — km) — vm[\ + k) 

V = M+^x ’ 

, _ MV(l + k') — v(m — k'M) 
v ~ M+m 


201. Only one body in motion, 
y, = V(M-km ) 

M+m ’ 


t — 


VM(l+k' ) 

M+m 


When a moving; body strikes a stationary elastic plane, its course of departure 
from the plane will be equal to its course of incident. 


N*®' 202 


2CL 


1 A Problem. A body in a is to strike the plane ABm 
® 7 that it will depart to the given noint b; required its 
} course of incident from a ? 

. Draw Id, at right angles through A13, make ed — 
i be join a and d; then ad is the course of incident, 
and eb. the course of departure, and the body will 
strike in e. 


Example 2. Fig. 197. The perfect elastic body M = 84 pounds, F= 18, m 
and v — 27. Required the velocity V' = ? after impact with the body m. 


48, 


V — 


18 (84 — 48) — 2X48X27 


= — 23-64. 


84+48 

the negative sign denotes that the body will return after the collision with a 
velocity of 23-63 feet per second. 

Example 3. Fig. 201. The partly elastic body M — 38 pounds and V — 79 feet 
per second, will strike the body in rest m = 24 pounds; what will be the velo¬ 
city v' — ? of the body m, its elasticity being lc’ = 0'6* 

, 79X38 a +0-6) . 

v' = -;--- ' — 70-6 feet per second. 

bo+24 


The bodies partly elastic. 



























154 


Friction. 


FRICTION. 


The resistance occasioned by Friction is independent of the velocity of mo* 
tion; but the re-effect of friction is proportional to the velocity. Friction is in- 
j dependent of the extent of surface in contact when the pressure remains the 
same, but proportional to the pressure. This law was established trom experi- 
| nients by Arthur Morin in the years 1831-32 and 1833, from which a summary 
I is contained in the accompanying Table. 

Letters denote. 

a — Fibres of the woods are parallel to themselves, and to the direction of 
motion. 

b — Fibres at right-angles to fibres. 

c = Fibres vertical on the fibres which are parallel to the motion. 

d = Fibres parallel to themselves, but at right-angles to the motion, length 
by length. 

e = Fibres vertical, end to end. 

Example. A vessel of 800 tons is to be hauled up an inclined plane, which 
inclines 9° 40' from the horizon; the plane is of oak, and greased with tallow. 
What power is required to haul her up ? 

The coefficient for oak on oak with continued motion is f = 0*097, say 0*1, 
then, 

800Xsin.9° 40' = 800X0*16791 = 134*328 tons, 
the force required if there were no friction, and 

800Xcos.9° 40'XO-l = 800X0*9858X0*1 = 78*864 tons, 
the force required for the friction only, and 
134*328 
78*864 

213*192 tons, the force required to haul her up. 

The effect lost by friction in axle and bearings is expressed simply by the 
formula 

ft d Wn f _ Wd nf 
12*60 */30 » 

in which W = the weight of pressure in the bearing, d = diameter on which 
the friction acts in inches, n = number of revolutions per minute, and/ = co¬ 
efficient of friction from the Table. In common machinery kept in good order 
the coefficient of friction can be assumed to / = 0*005. then 


P = 


P = 


Wdn 
353 7- 


H = 


Wd 

1941500 


Example. The pressure on a steam-piston is 20000 pounds, and makes n — 40 
double strokes per minute. Required the friction in the shaft of d = 8 incher* ? 




20000 X8X40 

1941500 

Friction 


= 3*3 horses, the loss by friction. 

in Guides* 


W= pressure on the steam piston in pounds. 
<8 = stroke of piston in feet. 

I = length of connecting rod in feet. 

H= horse power of the friction. 

n W s n 


350000 J of—S* 


sample. The pressure on a steam piston being W= 30,000 pounds, stroke 
= 1 feet, length of connecting rod l = 7 feet, and making 50 revolutions per 
Required the horse power of the friction H --- ? 


S 

minute. 


H = 


30000X4X50 


350000^5X7'^ 


1*13 horses. 





















Friction*. 


155 


1 


TABLE OF FRICTION FOR PLANE SURFACES IN CONTACT. 


Kind of Materials in contact. 
Oak on Oak, 


Cast-iron on Oak, 
a a 

U u 

Wrought-iron on Oak 
66 66 

Wrought iron, together^ 
66 6 . 

a a 

66 66 

Wrought on cast-iron, 

66 66 

66 66 

66 66 

Cast-Iron on cast-iron, 

66 66 

66 66 

66 66 

Wrought-iron on brass, 


Cast-iron on brass, 

66 U 


Brass on brass, 

6 % 66 

66 U 

Steel on cast-iron, 



Lubricated 

GoefficielU in 


with. 

Motion. 

Starting. 

a 

0 

0*47 S 

0-625 


tallow 

0*097 

0-160 


laid 

0-067 


b 

0 

0-324 

0-540 

33 

unctuous 

0-143 

0-314 

33 

tallow 

0-083 

0-254 





S3 




d 

0 

0-336 

.... 

c 

0 

0-192 

0-271 

e 

o 

• • • • 

0-43 

a 

0 

0-400 

0-570 

33 

soap 

0-214 

.... 

33 

tallow 

0-078 

0-108 

?3 

0 

0-252 

.... 

33 

tallow 

0-078 

.... 

a 

0 

0-138 

0-137 

a 

unctuous 

0-177 

.... 

33 

tallow 

0*082 

.... 

33 

olive oil 

0-070 

0115 

a 

0 

0-194 

0194 

33 

unctuous 

0-18 

0-118 

33 

tallow 

0-103 

0-10 

33 

olive oil 

0-066 

0 100 

a 

water 

0-314 

0-314 

33 

soap 

0-197 

.... 

33 

tallow 

0100 

o-ioo 

33 

olive oil 

0-064 

.... 

a 

0 

0-172 

.... 

33 

unctuous 

0-160 

.... 

33 

tallow 

0-103 

.... 

33 

lard 

0"075 

.... 

33 

olive oil, 

0*078 

.... 

a 

0 

0-147 

. • • 

33 

unctuous 

0-132 


33 

tallow 

0'103 

.... 

33 

lard 

0-075 




rwiTfi 






a 

0 

0-201 

.... 

33 

unctuous 

0-134 

. • • • 

33 

olive oil 

0-053 

.... 

33 

0 

0-202 

.... 

33 

tallow 

0-105 

.... 

33 

lard 

0 081 

.... 

a 

olive oil 

0-079 

1 


FRICTION 

OF AXLES IN 

MOTION. 




Oil, Tali mo. <r Hug's Lard. 


Dry or slightly 

Supplied in the. 

The grease 

Designation of surface in 

greasy, or wet. 

ordinary 

continually 

contact. 


manner. 

running. 

Brass on Brass, - 


0079 

• • . • 

“ on cast-iron, 


0-072 

0049 

Iron on Brass, - 

0*251 

0-075 

0-054 

“ on cast-iron, 


0-075 

0-054 

Cast-iron on cast-iron, 

0-137 

0-075 

0-054 

“ on Brass, - 

0-194 

0-075 

0054 

Iron on lignum-vita?, 

0-188 

0-125 


Cast-iron on 

0.185 

0100 

0-092 

Lignum-vitae on cast-iron. 


0116 

0170 





































156 


Strength of Materials. 


STRENGTH OF MATERIALS. 

Table I., shows the weight a column can bear with safety; when the weight 
presses through the length of the column. The tabular number is the weight 
in pounds or tons per square inch on the transverse section of a column of 
a length less than 12 times its smallest thickness. 


Table I. 

RESISTANCE FOR COMPRESSION. 


204 


Kind of Materials. 

Oak, of good quality, 

Oak, common, 

Spruce, red (Sapin rouge), 

“ white, (Sapin blanc 
Iron, wrought, 

Iron, cast, 

Basalt, 

Granite, hard, 

“ common, - 
Marble, hard, - 
“ common, - 
Sandstone, hard, - 
“ loose, - 
Brick, good quality, 

“ common, 

Lime-stone, of hardest kind, 

“ common, 

Plaster-Paris, - 
Mortar, good quality, and 18 months old, 
Do. common, - 


Pounds. 

432 

280 

540 

140 

14400 

28750 

2875 

1000 

575 

1435 

431 
1295 

5-6 

175 

58 

720 

432 
86 
58 
36 



When the length or height of the column is more than 12 times its smallest 
thickness, divide the tabular weight by the corresponding number in this 
Table. 


Length X thick ness 

12 

18 

24 

30 

36 

42 

48 

54 

60 

Divide by 

1-2 

1-6 

2 

2-8 

4 

5 

6 

8 

12 


Example. A building which is to weigh 2000 tons is to be supported by piles 
of Sapin rouge Spruce 18 feet in length, and 12 inches diameter. How many piles 
are required to support the building ? 


122X0-785X0-241 

1-6 


17 tons, the weight which each pile can hear, 


and 


2000 

17 


= 118 piles. 


To Find the Cohesive Strength* 

Rule. —Multiply the cross-section of the materials in square inches by the 
tabular number in Table II., and the product is the cohesive strength. 

Example An iron-bar has a cross-section of 2-27 sq. in. How many tons are 
required to tear it asunder, and how many pounds can it bear with safety? 

English iron 2 27X25 = 56-75 tons, which will tear it asunder, and it will bear 
with safety 

2-27X14000 = 31780 pounds. 








































Strength of Materials. 157 


Table II. 

COHESIVE STRENGTH PER SQ. INCH OF CROSS-SECTTON. 


Just tear 

asunaer. 

With safety. 


Kind of Materials. 

Pounds. 

Tons. 

Pounds. 

Tons 


Cast Steel, ... 

13425b 

59-93 

33600 

14-98 

zuo 

Blistered Steel, 

133152 

59-43 

33300 

14-86 

— 1 

7 

Steel, Shear, ... 

128632 

56-97 

32160 

14-24 



Iron, Swedish bar, - 

65000 

29-2 

16260 

7-3 



“ Russian, 

59470 

26-7 

14900 

6-7 



“ English, 

5COOO 

25-0 

14000 

6-25 



“ common, over 2 in. sq., 

36000 

lb-60 

9000 

4-0 



“ sheet, parallel rolling, 

40000 

17-85 

10000 

4-46 



“ at right angles to roll, 

34400 

15-35 

8600 

3-S4 



Cast iron, good quality, - 

45000 

20-05 

11250 

5-00 



“ inferior, - 

18000 

8-03 

4500 

2-0 



Copper, cast, - 

32500 

14-37 

8130 

3-6 



“ rolled, 

61200 

27-2 

15300 

6-8 



Tin, cast, * 

5000 

2-23 

12500 

0-56 



Lead, cast, ... 

880 

0-356 

220 

0-09 



roiled^ - - • 

00 ZU 

1*4:0 

00 U 

U*o/ 



Platinum, wire. 

53000 

23-6 

13250 

5-9 



Braes, common, 

45000 

20-05 

11250 

5-0 



Wood. 







Ash, .... 

16000 

7-14 

4000 

1-87 



Beach, - 

11500 

5-13 

2875 

1-28 



Box,. 

20000 

8-93 

5000 

2-23 


Cedar, .... 

11400 

5-09 

2850 

1-27 

/ \ 

Mahogany, ... 

21000 

9-38 

5250 

2-34 


“ Spanish, 

12000 

5-36 

3000 

1-34 


Oak, American white, - 

11500 

5-13 

2875 

1-28 

/ \ 

“ English “ 

10000 

4-46 

2500 

1T1 


“ seasoned,... 

13600 

6-07 

3400 

1-52 


Pine, pitch, 

12000 

5-35 

3000 

1-34 


“ Norway, - - - 

13000 

5-8 

3250 

1-45 


Walnut, .... 

7800 

3-48 

1950 

0-87 


Whalebone, ... 

7600 

3-40 

1900 

0-S5 


Hemp ropes, good, - 

6400 

2-86 

2130 

0-95 


Manilla ropes, - * 

3200 

1-43 

1100 

0-49 


Wire ropes, - - . 

38000 

17 

12600 

5-36 


Iron chain, ... 

65000 

29 

21600 

9-38 


“ with cross pieces, 

90000 

40 

30000 

13-4 



To Ascertain the Strength of Cables* 

Multiply the square of the circumference in inches by 120, and the product is 
the weight the cable will bear in pounds, with safety. 


Strength of Hemp Rope, with Safety. 


Circumference. 

Pounds. 

Circumference. 

Pounds. 

Circumference. 

Pounds 

1 - 

200- 

3’i 

2450- 

6 - 

7200- 

l*i 

312-5 

3-1 

2812-5 

6 -i 

7812-5 

1 -* 

450- 

4- 

3200- 

6 -i 

8450- 

1*1 

612-5 

4‘£ 

3612-5 

6-3 

9112-5 

2 - 

ere* 

4's- 

4050‘ 

7- 

9800- 

2 -i 

1012-5 

4-i 

4512-5 

7-i 

10512-5 

n 

1250- 

5* 

5000- 

Ti 

11250- 

n 

1512-5 

6 ‘* 

5512-5 

7-i 

12012-5 

3‘ 

1800- 

5*i 

6050* 

8 ‘ 

12800- 


2112-5 

5-3 

6612-5 




14 

























158 


Strength op Cables. 


CABLES. 

Strength of good Hemp Cable, with safety. 


mference. 

Pounds. 

Circumference. 

Pounds. 

Circumference. 

Pounds. 

6 - 

4320- 

10-25 

12607-5 

14-50 

25230" 

6-25 

4687-5 

10-50 

13230- 

14-75 

26107-5 

6-50 

5070- 

10-75 

13867-5 

- 15-• 

27000- 

6-75 

5467-5 

11 - 

14520- 

15-25 

27907-5 

7- 

5880- 

11-25 

15187-5 

15-50 

28830- 

7-25 

6307-5 

11-50 

15870- 

15-75 

29767-5 

7-50 

6750- 

11-75 

16567-5 

16- 

30720- 

7-75 

7207-5 

12 - 

17280- 

16-25 

31687-5 

8 - 

76S0- 

12-25 

18007-5 

16-50 

32670- 

8-25 

8167-5 

12-50 

18750- 

16-75 

33667-5 

8-50 

8670- 

12-75 

19507-5 

17- 

34680- 

8-75 

9187-5 

13- 

20280- 

17-25 

35707-5 

9- 

9720- 

13-25 

21067-5 

17-50 

36750- 

9-25 

10267-5 

13-50 

21870- 

17-75 

37807-5 

9-50 

10830- 

13-75 

22(587*5 

18- 

38880- 

9-75 

10 - 

11407-5 

12000 - 

14 

14-25 

23520- 

24367-5 

18-25 

39967-5 


Strength and Weight of Chains and Ropes■ 



Iron Chains. 

Hemp Ropes 

■j uimale Strength 

Absolute 

Strength 

Diameter 

Wight per 

Circumfer. 

Weight per 

Pounds. 

Tons. 

Pounds. 

Tons. 

in inches. 

foot, lb. 

in inches. 

foot, lb. 



JL 

8 

0-17 

li 

6-07 

920 

0-41 

306 

0.14 

3 

TS 

0-38 

2 

0-18 

2070 

0-92 

690 

0-31 

1 

4 

0-67 

2 -i 

2 

0-25 

3655 

1-63 

1218 

0-54 

5 

T 6 

1*08 

3 

0-36 

5720 

2-55 

1906 

0-85 

3 

IS" 

1-55 

3 £ 

0-56 

8200 

3-66 

2730 

1-25 

7 

16 

2-11 


0-72 

11100 

4-95 

3700 

1-65 

1 

2 

2-7 

5 

1 - 

14220 

6-36 

4740 

2-12 

JL 

1 6 

3'42 


1-32 

1 S 000 

8 - 

6000 

2-68 

5 

8 

4 

•1 

1-69 

22200 

9-9 

7400 

3-33 

1 1 

1 6 

4-84 

7 1 

7 4 

2-1 

26000 

11-6 

8960 

4 

3 

4 

5-75 

8 

2-33 

30000 

13-4 

10000 

4-5 

1 3 

Te 

6 

CO T?. 

CO 

2-84 

34000 

15-2 

11300 

5 

7 

¥ 

1 5 
16 

7-83 

9i 

3-30 

38000 

16-9 

12660 

5-6 

9-4 

10 

4-16 

41000 

18-3 

13700 

61 

1 inch. 

10-7 

1 °T 

4-6 

44800 

20 

15000 

6-7 


To ascertain the weight of Cable-laid Hopes. 

Multiply the square of the circumference in inches hy *036, and the product 
is the weight in pounds of a foot in length. 

To ascertain the Weight of Tarred Ropes and Cables. 

Multiply the square of the circtimference hy 2’13, and divide hy 9; the pro¬ 
duct is the weight of a fathom in pounds. 

Or, multiply the square of the circumference hy -04, and the product is the 
weight of a foot. 

For the ultimate, strength , divide the square of the circumference in inches hy 
5 ; the product is the weight in tons. 























Cables and Anchors. 


159 


CABLES AND ANCHORS. 


Table showing the size of Cables and Anchors proportioned to the Tonnage of 

Vessels. 


Tonnage oj 
Vessels. 

Cables. 

C iain Caoleo 

D, oof 

Weight oj 

Wright of' Weignt of a 

Circumference 

Diameter in 

in 

Anchor in 

fathom of 

fathom of 

in inches. 

inches. 

tons. 

pounds. 

Chain. 

Cables. 

5 

3- 

. 5 

*f 

56 

5-4 

2*1 



1 g 




8 

4- 

. 3 

8 

. 7 

l'f 

84 

8* 

4* 

10 


2-4 

112 

11* 

4*6 


1 g 



15 

5*4 

. i 

Y 

4- 

1C8 

14* 

6*5 

25 

6- 

. 9 

5. 

224 

17* 

8*4 



1 g 





40 

6t 

• 5 

g - 

6- 

336 

24* 

9*8 

60 

7- 

.1 1 

7- 

392 

27* 

11*4 



1 g 



75 

7*4 

. 3 

9' 

532 

30* 

13* 



T 




100 

8* 

.1 3 

1 g 

. 7 

8 

. 1 5 
lg 

1* 

10- 

616 

36* 

15* 

130 

9- 

12- 

700 

42* 

18*9 

150 

9-4 

14- 

840 

50* 

21* 

180 

10 4 

16- 

952 

56* 

25-7 

200 

11* 

I • 1 

1 ) 5 

18* 

117A 




60* 

28*2 

240 

12* 

i* 1 

V 

20- 

1400 

70* 

33*6 

270 

12't 

I’* 

21* 

1456 

78* 

36*4 

320 

13-4 

*•4 

22-4 

16S0 

86* 

42*5 

360 

14- 

1 ‘ rg 

25* 

1904 

96* 

45*7 

400 

14'i 

>•! 

27* 

2072 

104* 

49* 

440 

15-4 

1 *rg 

30* 

2240 

115* 

56* 

480 

16' 

v i 

33* 

2408 

125* 

59*5 

620 

16-4 

1 ’rV 

36* 

2800 

136* 

63*4 

570 

17* 

i* 5 

31* 

3360 

144* 

67*2 

620 

17-4 

l‘Ii 

1 1 G 

42* 

3920 

152* 

71*1 

680 

18- 

4* 

45* 

4200 

161. 

75*6 

740 

19- 

1 *Tg 

49* 

4480 

172* 

84*2 

820 

20- 


52* 

5600 

184* 

93*3 

900 

22- 

1 1 6 

56* 

6720 

196* 

112*9 

1000 

24- 

2* 

60* 

7168 

208* 

134*6 


The proof in the U. S. Naval Service is about 12| per cent, less than the above 
for the larger sizes, and from 25 to 30 per cent, for the smaller. 

The results of experiments at the U. S. Navy Yard, Washington, D. C., give 
for the cohesive force of chain iron, per square inch, as follows: 


Mean of experiments with good iron, - -- -- -- -- - 41000 lbs. 
Mean of experiments with best iron,.. 46000 lbs. 



















160 


Weight and Shrinking of Castings, 


To find the weight of Castings* hy the weight of 
Pine Patterns* 


Multiply the weight 
of the Pattern by 


RULE. 

( 12 for Cast Iron, 

\ 13 “ Brass, 

< 19 “ Lead, 

I 12-2 “ Tin, 

V. 11-4 “ Zinc, 


( and the product is the 
l weight of the Castings. 


Seductions for Round Cores and Core-prints. 

Ride. Multiply the square of the diameter by the length of the Core in 
inches, and the product by 0-017, is the weight of the pine core, to be deduc¬ 
ted from the weight of the pattern. 


Shrinking of Castings-. 


Pattern Makers’ Rule \ 

Cast Iron, 
Brass, 

Lead, 

i inch 

f longer per Linear 

should be for j 

Tin, 

A a ' 

Foot. 

( 

Ziuc, 

3 o . 

I 6 

) 


Weight and Capacity of Balls* 

Diameter in 

Capacity in cubic 

CAST IRON. 

LEAD. 

inches. 

inches. 

Pounds. 

Pounds. 

| I ’ 

•5235 

•1365 

•2147 


1-7671 

-4607 

•7248 

2 - 

4-1887 

1-0920 

1-7180 

2-4 

8-1812 

2-1328 

3-3554 

3 - 

14-1371 

3-6855 

5-7982 

3-4 

22-4492 

5-8525 

9-2073 

4 • 

33-5103 

8-7361 

13-744 

4-4 

47-7129 

12-4387 

19-569 

5 - 

65-4498 

17-0628 

26-843 

54 - 

87-1137 

22-7206 

35-729 

6 - 

113-0973 

29-4845 

46-385 

6-4 

143-7932 

37-4528 

58-976 

7 - 

179-5943 

46 8203 

73-659 

7-4 

220-8932 

57-5870 

90-598 

8 - 

268 0825 

69-8892 

109-952 

8-4 

321-5550 

83-8396 

131-383 

9 - 

381-7034 

99-5103 

156-553 

9-4 

448-9204 

117-0338 

184-121 

10 - 

523-5987 

136-5025 

214-749 

11 * 

696-9098 

181-7648 

285-832 

12 - 

904-7784 

235-8763 

371-096 

13 - 

1150-346 

299-6230 

471-806 

14 - 

1436-754 

374-5629 

589-273 

15 - 

1767-145 

460-6959 

724-781 

16 - 

2144-660 

559-1142 

879-616 

17 * 

2572-4 0 

670-7168 

1055-066 

18 - 

3053-627 

796-0825 

1252-422 

19 - 

3591*363 

936-2708 

1472-970 

20 - 

4188-790 

1092-0200 

1717-995 
































To Find the >“solute Strength. 


161 


TO FIND THE LATERAL STRENGTH. 

7 Rule. —Find the coefficients k and x for the material on page 164. Insert this 
m the formula for the corresponding section of the materials, and the formula 
gives the absolute strength in pounds. 


Expressions^^" 


and t — inches. 


Example 1. Fig. 208. A rectangular beam of oak fastened in a wall, projects out 
1=6 feet, 4 inches, h = 8 inches, and 6 = 5 inches. How much weight can it 
bear on the end? 


W = 


30X5X8° 

6-333 


1509 pounds, with perfect safety. 


Example 2. Fig. 209. A beam of the section T, supported at each end is 
l — 8'75 feet; dimensions of the section are h = 10, 6 = 4, c = 1-25 and t = 1-5 
inches. The beam is of cast iron, for which k = 160 and x = 2000. Required 
how much weight can it bear on the middle VF=? and how much at n. = 3 feet 
from one end TV' = ? 

4 4 

W — — \k(t ft°xo-46 e*)+z 6 h e] = — [150(l-5Xl0 a +0-4X44Xl-25*)+1000 
l o*7 o 


X4X10X1*25] = 
and 


35976 pounds = 16 tons on the middle with perfect safety, 


W f = 


TV l* 35976X8-75° 

4m n 4X5‘75X3 


39919 pounds = 17.8 tons, 


at 3 feet from the end. 

The coefficient k here given is about one fourth of the extreme strength at 
which the material would break, which is considered the practical value when 
the force or load is acting continually for a term of years. When the force is 
acting at intervals, and greater per centage is desired, | of the extreme strength 
can be relied upon, or £, if circumstances are pressing, but the latter never to be 
exceeded. 

The weight of the materials itself has not here been taken into consideration, 
for which allowance must be made, if considerable. 



206. 


A beam fixed in one end and loaded at the 
other, should have the form of a Parabola in 
which 

l = abscissa, and 
h = ordinate. 



207. 

To cut out the stoutest rectangular learn from a 
log. 

1st case, divide the diameter in 3 equal parts, 
and draw lines at right-angles as represented. 
2nd, divide the diameter in 4 equal parts. 

1 , l.i = 1-414 6 , non-elastic. 

2 , h = 1'73 6 , elastic beams. 


* 































162 Strength of Materials, with Safety. 













































Strength op Materials, with Safety. 


163 






















































164 


Strength of Materials, •with Safety 


Coefficients for t/lve Lateral Strength. 


Transverse Suction 

ill v 

Chsi i?*o«. 

A - 150. 

Wrought Iron. 

A = 120. 

Oa/'r or Pine. 

A = 30. 


A: - 150. 

A A 3 = s\ 

A = 120. 

A 3 A = 5 3 . 

A = 30. 

A 2 A = 5* 





A = 88. 

A 3 A = d*. 

A = 70. 

A 3 A = <F. 

A = 18. 

A 3 A = rf*. 

i 

A = 88. 

h? b = D* — • d *- 

A = 70. 

A 3 A 3 = D 3 — d z . 

A = 18. 

A 8 A = 7)* — d* 

'mmm$ 

A = 150. 

* = 1000. 

A = 120. 

x = 1500. 

A = 30. • 

a: = 400. 

Y/fi. 

-pfb * 

W 

mr I 

A = 144. 

A = or > 3e. 

A = or > 3£. 

A - 115. 

A = or > 3e. 

A = or > 3^ 

A = 29. 

A = or > 3e. 

A *= or > 3f. 

extension. 




v J' if 

jyl 

A = 150. 
a? = 1000. 

A = 5c. 

A = or > 2 A. 

A = 120. 
a? = 1500. 

A = 5c 

A = or >2 A. 

A = 30. 
x - 400. 

A = 5c. 

A = or > 2A. 

compression. 





compression. 
' A 

w- 

extension. 


The coefficients and proportions are the same as the next 
above, but the side marked extension must be 
up in the Figs. 208. 210. 212. 214. 
down in the Figs. 209. 213. 



















































Strength of Materials. 


165 


ABSOLUTE AND 

Table III. 

EXTREME STRENGTH OP MATERIALS. 



Coefficient lc. 


Kind of Materials. 

Continually 

At 

Pressing 

Extreme. 


for years. 

intervals. 

Circumstances 

Break asunder. 

Iron wrought, - - - 

120 

162 

240 

488 

“ cast,. 

150 

200 

300 

600 

Steel, soft,. 

175 

233 

350 

700 

Brass, -. 

58 

75 

113 

226 

Copper,. 

53 

71 

106 

212 

Zinc, -'--I.-- 

15 

20 

30 

61 

Tin,. 

17 

23 

34 

69 

Lead, ------- 

4 

6 

9 

18 

Ash,. 

45 

56 

85 

170 

Hickory,. 

G7 

90 

135 

270 

Chestnut, sweet. - - - 

42 

56. 

85 

170 


50 

66 

100 

200 

“ English, - - - - 

25 

33 

50 

loo 

“ Canadian, - - - 

37 

49 

73 

147 

Pine, white, ... - 

34 

45 

67 

135 

“ yellow, - - - - 

38 

50 

75 

150 

Teak,. 

51 

68 

102 

205 

- n - 


BRICKS. 


Dimensions. 

Common brick 8X44X2^ inches = 85 cubic inches. 
Front brick 84 X 4 ^X 24 „ = 92"8 „ „ 

When laid in a wall with cement it occupies a space of 
Common brick 84X44X2f inches = 102 cubic inches. 
Front brick 8iX4fX2f „ -111 „ „ 

Wtii“HI and Bulk of Bricks* 


Tons. 

Pounds. 

Cub. feet. 

Number 

by itself. 

C. Brick. P. Brick. 

of bricks 

in wall wil 
C. Brick. 

;h cement 
F. Briek. 

l 

2240 

22-4 

448 

416-6 

381 

347 

0-04464 

100 

1 

20 

18-6 

17 

15i 

2-23 

5000 

50-00 

lOOO 

930 

850 

772 

2-4 

5376 

53-76 

1075 

lOOO 

914 

834 

2-62 

5872 

58-72 

1130 

1100 

lOOO 

913 

2-88 

6451 

64-51 

1240 

1200 

1100 

lOOO 


•-► «- 

WOOD. 

A Cord of wood is 4 feet wide, 4 feet high, and 8 feet deep. 
128 cubic feet. 













































166 


Gearing. 


GEARING. 



Letters denote. 

P = pitch,—the distances between the centres of two teeth in the 
pitch circle. 

D — diameter 

C = circumference ( of the wheel. 

M — number of teeth £ 

N — number of revolutions ' 
d — diameter -\ 

c = circumference t 

m — number of teeth f 0 e P in1011, 

n = number of revolutions J 



Gircivm. - 

<7= PM - 5 

C=rt D -6 

Diameter - 

,V= PM - 7 

c 

lj)= - . 8 

Tt 


D : d = C : c — M: m —n : N 
Example 1. A wheel of D — 40 inches in diameter, is to have M — 
75 teeth. Required the pitch P — ? 

Formula 2. Pitch P — — 3 14X40 —_ ^.gg inghgg nearly. 

75 

Example 2. The pitch of teeth in a wheel, is to be P — 1*71 inches, and 
having M — 48 teeth. Required the diameter D = ? of the wheel. 

^ == 26 , 14 in. of the pitch circle 

3*14 


Formula 7. Diam. D 






















Gearing. 


167 


Construction of Teeth for Wheels* 


Draw the radius R r, and pitch circle P P. Through r draw 
an angle of 75° to the radius R r. 

the line off at 

Half the angle be¬ 
tween two teeth in the 

, i 180 

wheel, v = ——. 

M 

.. _ iso; 

L pinion, V =—™ 

• 

- 1 

- 2 

D 

: d = sin. V: sin. v. 


• 

Diameter of the < 

wheel, D = dsin - V . 

sm. v 

, D sin. v 

'P lmon >'*“ dn _ y ■ 

“ 

- 3 

- 4 

Pitch (chord) of teeth J wheel* P = D sin. v. 
in the pitch circle l pinion, P = d sin. V. 

• 

- 5 

- 6 

Approximate pitch in the wheel P — 0'028 D. - 

- 7 


, t 3T4D 

/- wheel, M = ——— - 

- 

- 8 

Number of teeth about 

1 . . dM 

v pinion, m ~~ • — jj —* - 

- 

- 9 

Thickness of tooth, a = 0*46 P* - 

- 

10 

Bottom clearance* b 

= 0‘4 P. 

- 

- 11 

Depth to pitch line, 

c = 0‘3 P. 

- 

- 12 

Distance r o, d = 

2 (m — 11) 

- 

- 13 

Distance r o’, e = 0T1 P l/m 

- 

. 14 

* If a wheel of more than 80 teeth is to gear a pinion of less than 20 teeth, 
and the wheel and pinion are of the same kind of materials ; take the thickness 


wheel, a = P (o-42 + 

m \ 

fW) 

m 

1 - - 15 

of the tooth in the •< 

V 

pinion 0-5 p(l- 

). - 16 



' 350 


A rack is to be considered as a wheel of 200 teeth. 









168 


Gearing. 


Example with Plate 1V« 

Example. A •wheel of D = 48 inches diameter is to gear a pinion about 8 
revolutions to 1. Required a complete construction of the gearing? 


Approximate pitch P = 0*028X48 = 1’34 in. 


Number of teeth 
in the 


wheel, M — 


3-14X48 

1'34 

112 


= 112 . 


Half the an- | 
gle between -< 
two teeth in ( 
the 


pinion, m a 
wheel, v == 


8 


= 14 


=1°36'. sm=0'028. 

LIZ 


pinion T r = 

Diameter of pinion d 


180 


= 12°51'. sm= 0'2 2 24. 


14 

48X0-028 

0-2224 


= 6*043 in. 


Draw the pitch circle for the wheel and pinion so that they tangent one an 
other at r on a straight line between the centres of the circles. 

Pitch in the gearing P= 48X0'028=T*344 in. - 5 

Take this chordial pitch in a pair of compasses, and set it off in the pitch 
circles. 


Thickness of J 
tooth 


f Wlieci ” ” A ™ ( Q,/ 


wheel a ~ 1'344f 0*42 + 


_U 

700 

14 


) 


—0*592in. 15 


i . . / 14 v 

'•pinion «= 0'5 1*344? 1—g^-1=0*645 in. 


Set off the thickness of tooth in the corresponding pitch circles. 

Bottom clearance b =* 0*4X1-344 = 0'5376 in. 
Depth to pitch line c = 0*3 1*344=0*4032 in. 


Distances r o and 
r o' in the wdieel 


, 1*344(112 6) , 7Q ,, . 

d _ Xu^-ii ) = 78 ia * 


16 

11 

12 

13 


e = 0*11X1*344 v/112 = 0'7120in, 14 

Set off these distances on the line o o' from r, — d beyond and e within the 
pitch circle for the wheel \ then o is the centre and o m radius for the flank m. 
o' the centre and o' n radius for the face n. Draw circles through o and o' con¬ 
centric with the pitch circle of the wheel. 


Distances r o and 
r o' in the pinion 


d 


P.344(14+ 6J = . n< 

2(14—11) 

e = 0*11X1*344 v' fi = 0*356 in. 

Proceed with the pinion similar as the wheel 

On the plate is a scale of inches and decimals, which will be con¬ 
venient for the above measurements. 


13 

14 





















^ ^ 


Slate. n: 


J. WNy.'strain. 





































































































































Tv.. . — r 


v r ■. 5 «. 


v. c 














































ijnrj-qms' jo a Ml 


Jlat& V. 


v\ 







Journals iJuuncta: 


J.W.JVystrom. 







































































































































































































































































































































































































































■ -- ■ V’ •• y** - 






- ■* •* 






























N 
























■ 






































■ • • I ; 1 

! f ■< ' 

: y ■- » • 












■': : : • • 1 ■ • • ^ • 
"V-:^ V * ' 

* 

v • i • r , ■■. * * <-r > . 




















■ 


. r. V 

1 i.. s’ ,. • 

































































■ 


■ 


,s • 




























. 




>■- - • 






- .y > ■ - -•* 




















» 








Strength op Materials. 


169 


Strength of Teeth* 

Letters denote. 

S — strain on the teeth at the pitch-line in pounds. 
a = thickness, (see figure), \ . . 

h = breadth of the teeth, j 
v = velocity of the teeth in feet per second. 

H — Horse power transmitted by the teeth. 


r a — 0'025|/ S, 

Thickness 

l a = 


Pitch 



{ S = 1600 a\ 
S = S00 h a. 


Horses 


{ 


H= 2-275 Vav, 


H — 0'48}/j3 v. 


Abrasion included in these formulas and the breadth h — 2-5 p. 

When great strain is required on the teeth, and the diameter or pitch of the 
wheel is limited, it is necessary to increase the breadth h proportionally, which 
will be thus. 


h = 


S_ 
653 p 


S 

300a 


Sm 
943 D 


Tim 


H 


1-347 D v 0'429a v 0'934p v’ 


Example. The pinion-wheel on a propeller shaft is to be D = 48 inches in 
diameter, and to have m = 36 teeth; it is driven by a pair of engines H = 450 
horses, and the propeller to make n — 50 revolutions per minute. Required 
the breadth of the teeth h = ? The velocity at pitch circle will be, 

3-14X4X50 _ . . . 

— 10-5 feet per second, nearly, and 


h = 


v — 


IT m 


60 


450X36 


1-347 Dv 1-347X^X10-5 


- = 23-88, or 24 inches, nearly. 


To Find the Diameter of Axles and Shaft, 

Letters denote, 

d — diameter in inches, in the bearing; and the length of the bearing 1-5 d. 
W = weight in pounds, acting in the bearing. 


Water¬ 

wheels 


d 


yw 

n 

I fw 


of cast iron. 


of wrought iron. 


24 


Common 
Machinery J 
in good ] ,/«> 

order. 


d — X~- of cast iron. 


i yw 

d = — of wrought iron. 


Example 1. A water wheel weighs 58,680 pounds, and supported in two bear¬ 
ings. Required the diameter of the wheel axles? The weight acting in each 
bearing will be 58680 : 2 = 29340 pounds, and 

diameter d = ^ 1! ~~~~~ = 8* 15 inches oi wrought iron. 

L 1 

Example 2. Fig. 226, page 185. Required the diameter of the axle in the 
wheel, when the weights P +, : Q = 4864 pounds? If the wheel is supported 
in two bearings W = 4864 : 2 = 2432 pounds. 

'" 24‘>2 

diameter d = ^ -— = 1'76 inches of wrought iron. 

28 


15 




















170 


Strength of Materials. 


Example 3. The pressure on the steam piston, in a walking beam engine is 
25000 pounds. Required the diameter of the beam journals 1 

diameter d — ~ 00 ^^ ' = 5*64 inches the eentre one. 

Zo 


d — — ^ ^ inches at the ends. 

In this example it is supposed that the beam is worked by a fork connecting 
rod. 





y f r 


77 

Vir 


D = inches wrought iron. 

R = radius of crank in feet. 

F = force from the steam piston, lbs. 



D : d — y/ R : \7 r 
J)= 4-ss* ~ f H 


h35 */ H 
v n 


H — horse-power transmitted. 
n = number revolutions per minute. 


When an axle or shaft not only serves as a fulcrum, but effect is transmitted 
by the act of twisting it, the diameter is to be caluulated as follow. 

Example 4. The pressure on the piston in a steam engine is F — 45,600 
pounds, applied direct on a crank of R = 3 feet radius. Required the diameter 
of the shaft and crank pin ? 

Diameter of the shaft D = V 4o<.i()()*8_ 12*9 inches. 

4 


Diameter of the crank pin d 


= V 45600 
28 


7*63 inches. 


Example 5. A steam engine of 368 horses is to make 32 revolutions per 
minute. Required the diameter of the main shaft? 


Diameter D = 



= 11 { inches. 


Example 6. A cog wheel of R = 6-5 feet radius is to gear with a pinion of 
r = T25 feet radius, and to transmit an effect of 231 horses with 42 revolutions 
per minute. Required the diameter of the wheel aud piniou shafts ? The force 
Fis acting uniformly at the periphery. 


3 / 231 

V 42 “ 
7 r 

Diameter of pinion shaft d — 7*66 a / J.J; 

V 6 * 


Diameter of wheel shaft D = 4*35 
D:d = 


= 7*66 inches. 


4*41 inches. 


































Allots. 


171 


ALLOYS. 

Letters denote. 

A — Antimony, B = Bismuth, C — Copper, G — Gold, I — Iron, L — Lead, 
N = Nickel, S = Silver, T — Tin, and Z = Zinc. 


Brass, yellow, ... 

“ rolled, - 

Brass-casting, common, 

“ hard, - 

Brass-Propellers, (large), - 
Gun-metal, .... 
Copper-flanges, for pipes, - 
Brass that bears soldering well, 


Muntz’s metal can be rolled and worked at red heat, 6(7, 4 Z. 


2(7, 1 Z. 

32(7, 10Z, 1-57’. 
206; 1-25Z, 2-521 
25(7, 2Z, 4-5 T. 
8(7, 0-5Z, 12’. 

8(7, IT. 

9(7, 1Z, 0-2621 
2(7, 0-75Z. 


Statuary, 

German Silver, - 
Frick’s Imitative Silver, 
Medals, - - - 

Pinchbeck, ... 
Chinese Silver, - 


91-4(7, 5-53Z, 1-72 7 ,1-372:. 
20(7, 15-82V, 12-7Z, 1-3/. 
53-39(7, 17-4A7 13Z. 

100(7, 8 Z. 

5C, 1Z. 

65-2(7, 19-5Z, 132V, 2-5 S, 12 
cobalt of I. 

Britannia metal, .... - 1Z, Id ).» n 

When f used add, .Id, IB f 

Babbitt’s anti-Attrition metal, .... 252*, 2d 0-5(7. 

The Tin of the best quality of Banca, is to be added gradually to the melted compo¬ 
sition. 

Bell-metal, large, ...... 3(7, 121 

u small, ...... 4(7,121 

Gold Metal. 

* = 6^(7 9 me ^ te( l separately. 

Gold = 71y+9a:, this makes a brilliant composition. 


Solders. 


Newton’s fhsible alloys, 

Rose’s “ “ 

A more fusible composition 
Tin solder, coarse, 

“ ordinary. 

Soft Spelter-solder, for common 
Hard “ for iron, 
Solder for Steel, 

Solder for fine brass works, 

Pewteret’s soft solder, - 
a « 

Gold Solder, 

Silver solder, hard, 

“ soft, 


8 JB, 5 L, 3T, melts at 212 °. 


2 B, 1 L, IT, 

5 R, 3 L, 2T, 

IT, 3 L, 

22’, IZr, 

brass works , 1(7, 1Z. 

2(7,1Z 
19N, 3(7, 1Z. 

IN, 8(7, 8 Z. 

2B, 4 L, 32: 

15,15, 27’ 

24 G, 2 S. 1C. 

IS, 1(7. 

- 2S, 1 brass wire. 


201 o. 

199°. 

500°. 

360°- 


Tempering Steel, 

Tem. Fah. 

Yellow, very faint, for lancets ..... 430° 

„ pale straw, for razors scalpels .... 450 ° 

„ full, for penknives and chisels for hard cast iron - 470° 

Brown, for scissors and chissels for wrought iron - - 490° 

Red, for carpenter tools in general - - * 510° 

Purple, for fine watch springs and table knives - - 530° 

Blue, bright, for swords, lock springs .... 5500 

„ full, for daggers, fine saws, needles - - 560° 

„ dark, for common saws ..... 600° 


L 

















172 


Weight of Rolled Iron, per Foot. 



Side ID 
inc'ie<). 

Welsh' in 
pounds. 

tide ia 
inches. 

Weight in 
p unds 

iV 

0-013 

3# 

44-418 

1 

0*53 

3f 

47-534 

S 

1 ^ 

0-118 

31 

50-756 

1 

0-211 

4 

54-084 

t 

0-475 

41 

57-517 

1 

0-845 

41 

61-055 

§ 

1-320 

4§ 

64-700 

£ 

4 

1-901 

41 

68-448 

1 

2-588 

4f 

72-305 

1 

3-380 

4S 

76-264 

H 

4-278 

41 

80-333 

li 

5-280 

5 

84-480 

if 

6-390 

51 

88-784 

H 

7-604 

51 

93-168 

if 

8-926 

5f 

97-657 

U 

10-325 

51 

102-24 

11 

11-883 

5$ 

106-95 

2 

13-520 

51 

111-75 

21 

15-263 

51 

116-67 

21 

17-112 

6 

121-66 

2# 

19-066 

61 

132-04 

21 

21-120 

61 

142-82 

2f 

23-292 

6f 

154-01 

2f 

25-56 

7 

165-63 

2f 

27-939 

71 

190-14 

3 

30-416 

8 

216-34 

31 

33-010 

81 

244-22 

31 

35-704 

9 

273-79 

3f 

38-503 

10 

337-92 

31 

41-408 

12 

486-66 



iJianie'.er 
in incner.. 

Weiytit in 

pounds. 

1 'idmeter 

in indies. 

VVeight in 

pounds. 

tV 

0-010 

31 

34-886 

1 

0-041 

3i 

37*332 

? 

Iff 

0-119 

31 

39-864 


0165 

4 

42 464 

t 

0-373 

41 

45-174 

1 

0-663 

41 

47-952 

1 

1-043 

4f 

50-815 

1 

1-493 

41 

53-760 

1 

2-032 

4f 

56-788 

1 

2-654 

4i 

59-900 

H 

3-360 

41 

63-094 

li 

4-172 

5 

66 752 

If 

5-019 

51 

69-731 

11 

5-972 

51 

73-172 

If 

7-010 

5f 

76-700 

li 

8-128 

51 

80-304 

U 

9-333 

5f 

84-001 

2 

10-616 

5i 

87-776 

21 

11-988 

51 

91-634 

21 

13-440 

6 

95-552 

2f 

14-975 

61 

103-70 

21 

16-688 

61 

112-16 

2f 

18-293 

6 ! 

120-96 

21 

20-076 

7 

130-05 

21 

21-944 

71 

149-33 

3 

23-888 

8 

169-85 

31 

25-926 

81 

191-81 

31 

28-040 

9 

215-04 

3f 

30-240 

10 

266-29 

31 

32-512 

12 

382-21 


Cements for Cast Iron* 

Two ounces Sal-ammoniac, one ounce Sulphur and sixteen ounces of 
borings or filings of cast Iron, to be mixed well in a mortar, and kept dry. 
When required for use take one part of this powder to twenty parts of clear 
iron borings or filings, mixed throughly in a mortar, make the mixture into a 
stiff paste with a little water, and then it is ready for use. A little fine grindstone 
sand improves the cement. 

Or one ounce of Sal-ammoniac to one hundred weight of Iron borings. No 
heat allowed to it. 

The Cubic contents of the joint in inches, divided by five, is the weight of dry 
borings in pounds Avoir, required to make cement to fill the joint nearly. 

Cement for Stone and Brick work* 

Two parts Ashes, three of Clay, and one of Sand, mixed with oil, will resist 
weather equal to marble. 

Brown Mortar* 

One part Thomaston lime, two of Sand, and a small quantity of Hair. 

Hydraulic Mortar* 

Three parts of Lime, four Puzzolana, one Smithy Ashes, two of Sand, and four 
parts of rolled stone or shingles. 








































173 


Weight of Riveted Pipes, per Foot, the Laps included. 


Diainetei 

Thickness 

Copper pipe 

Iron pipe 

Diameter 

Thickness 

Copper pipe 

Iron pipe 

inches. 

16 th in. 

pounds. 

pounds. 

inches. 

16 th in. 

pounds. 


pounds. 

5 

3 

12-50 

10-96 

9* 

4 

30-59 


26-81 

5 

4 

16-88 

14-78 

10 

4 

32-21 


28-21 

5£ 

3 

13-15 

11-52 

10* 

4 

33-94 


29-70 

H 

4 

17-75 

15-55 

11 

4 

35-20 


30 84 

5* 

3 

13-63 

11-94 

11* 

4 

36-94 


32-35 

5* . 

4 

18-39 

16-07 

12 

4 

38-45 


33-67 


3 

14-25 

12-48 

13 

4 

41-45 


36 30 

5f 

4 

19-25 

16-86 

14 

4 

44-64 


39-11 

6 

3 

14-76 


12-94 

14 

5 

55-88 


43-97 

6 

4 ; 

19-91 

17*43 

15 

4 

47-64 


41-74 

6£ 

3 

15-36 

13-46 

15 

5 

59-59 


52-20 

6£ 

4 

20-75 

18-16 

16 

4 

50-75 


44-45 

6* 

3 

15-90 

13-93 

16 

5 

63-47 


55-60 

6* 

4 

21-41 

18-75 

17 

4 

53-86 


47-15 

61 

3 

16-50 

14-45 

17 

5 

67-34 


59-00 

62 

4 

22-25 

19-70 

18 

4 

57-04 


50-00 

7 

3 

17-03 

14-93 

18 

5 

71-26 


62-41 

7 

4 

22-93 

20-07 

19 

4 

60-14 


52-65 

7i 

3 

17-65 

15-45 

19 

5 

75-23 


65-90 

7i 

4 . 

23-74 

20*79 

20 

4 

62-51 


54-74 

7* 

3 

18-32 

16-05 

20 

5 

78-21 


68-5 

7* 

4 

24-45 

21-40 

21 

5 

82-98 


72-64 

74 

3 

18*95 

16-60 

22 

5 

86-77 


76-00 

74 

4 ■ 

25-28 

22-13 

23 

5 

90-57 


79-34 

8 

3 

29-42 

17-03 

24 

5 

94-31 


82-60 

8 

4 

25-96 

22-72 

26 

5 

101-9 


89-32 

8* 

3 

20-58 

18-03 

28 

5 

109-4 


95-68 

8* 

4 

27-47 

24-07 

30 

5 

117-0 


102-4 

9 

4 

28-98 

25-38 

36 

5 

140-0 


122-5 


Wei; 

;ht of Cast Iron Cylinders per 

Foot. 



Diameter 

Weight 

Ijiimtter 

Weight 

Diameter 

Weight 

Diameter 


Weight 

in 

»n 

in 

in 

in 

in 

in 


in 

inches. 

pounis. 

inches. 

pounds. 

inches. 

pounds. 

inches 


pounds. 

4 

1-39 

2} 

18-74 

4£ 

55-92 

7* 


139-4 

i 

1-88 

21 

20-48 

4£ 

58-72 

7f 


148-87 

lin. 

2-47 

3in. 

22-35 

5 in. 

61-96 

Sir). 


158-63 

li 

3*13 

H 

24-20 

5i 

64-66 

8* 


168-15 

H 

3-87 

3± 

26-18 

5£ 

68-31 

8* 


179-1 

U 

4-68 

3t 

28-23 

5f 

71-00 

8| 


189-0 

i* 

5-57 

3£ 

30-36 

5* 

74-98 

9in. 

200-8 

H 

6-54 

3f 

32-57 

5f 

78-65 

9* 

21112 

14 

7*59 

3| 

34-85 

51 

81-95 

9* 

223-7 

u 

8-71 

3£ 

37-21 

5| 

85-81 

9f 

235-3 

2m. 

9-91 

4m. 

39-66 

6m. 

89-23 

10lM. 

247-9 

li 

11-19 

4i 

41-80 - 

6* 

96-82 

10* 

273-27 

2i 

12-54 

4* 

44-77 

6* 

104-7 

11 in. 

299-9 

2£ 

13-98 

43 

47-00 

6f 

112-9 

11* 

327-8 

2* 

15-49 

4* 

50-19 

7in. 

112-4 

12in. 

356-9 

2% 

1708 

4f 

52-71 

7i 

130-28 

13in. 

418-9 

































174 Weight of Cast Iron Pipes per Foot. 


! Pore. 

f I J _ 

Thick 

* 

ness of Metal. 

i ! 1 

! S 

t 1 

1 

3.06 

1 5-05 




i 


H 

3-67 

6-00 






1* 

6-89 

6-89 

9-81 



1 


If 


7-80 

11-04 



( 


2 


8-74 

12-23 

16-0 




2± 


9-65 

13-48 

17-52 




1 2* 


10-57 

14-66 

19-05 

23-8 



2f 


11-54 

15-91 

20-59 

2568 



3 


12-28 

17-15 

22-15 

,27-56 

33-30 

39-31 

3* 


13-24 

18-40 

23-72 

29-64 

35-46 

41-77 

3* 


14-20 

19-66 

25-27 

31-20 

37-63 

44-23 

3f 


15-50 

20-90 

26-83 

33-07 

39-77 

46-68 

4 


16-80 

22-05 

28-28 

34-94 

41-92 

49.14 

4± 


17-41 

23-35 

29-85 

36-73 

44-05 

51-57 

4* 


18-00 

24-49 

31-40 

38-58 

46-19 

54-00 

4f 


18-89 

25-70 

32-91 

40-43 

48-34 

56-45 

5 


19-79 

26-94 

34-34 

42-28 

50-50 

58-90 

5* 


2154 

29-40 

37-44 

45-94 

54-81 

63-82 

6 


23-42 

31-82 

40-56 

49-60 

58-96 

68-70 

6* 


25-26 

34-32 

43-68 

53-30 

63-18 

73-40 

7 


27-15 

36-66 

46-80 

56-96 

67-60 

78-39 

7* 


28-92 

39-22 

49-92 

60-48 

71-76 

83-28 

8 


30-76 

41-64 

52-68 

64-27 

76-12 

88-20 

8* 


32-82 

44 11 

56-16 

68-00 

80-50 

73-28 

9 


34-45 

46-50 

58-92 

71 70 

84-70 

97-98 

9* 


36-26 

48-98 

62-02 

75-32 

88-98 

102-9 

10 


38-15 

54-46 

65-08 

78-99 

93*zi 

108-8 

10 * 



53-88 

68-14 

82-68 

97-44 

112-7 

11 



56-34 

71-19 

86-40 

101-83 

117-6 

n* 



58-82 

74-28 

90-06 

106-1 

122-6 

12 



61-26 

77-36 

93-70 

110-5 

127-4 

12* 



63-70 

80-40 

97-40 

114-7 

132-5 

13 



66"14 

83-46 

101-1 

118-9 

137-3 

13* 



68-64 

86-55 

104-8 

123-3 

137-28 

14 



71-07 

89-61 

108-46 

127-6 

147-0 

H* 



73-72 

92-66 

112-1 

131-9 

151-9 

15 



75-96 

95-72 

1L5-8 

136-2 

156-8 

15* 



78-40 

98-78 

119-5 

140-4 

161-8 

16 



80-87 

101-8 

123-1 

144-8 

166-6 

16* 



83-30 

104-8 

126-8 

149-0 

171*6 

17 



85-73 

107-9 

130-5 

135-3 

176-6 

17* 



88-23 

111-1 

134-2 

157-6 

181-3 

18 




114-1 

137-8 

169-9 

186-2 

19 




120-2 

145-2 

170-5 

195-9 

20 




126-3 

152-5 

179-0 

205-8 

21 




132-5 

159-8 

187-6 

215-5 

22 




138-6 

167-2 

196-5 

225-4 

23 




144-8 

147-6 

204-8 

235-3 

24 




150-8 

181-9 

213-28 

245-1 

26 





196-6 

230-6 

264-7 

28 





211-3 

247-6 

284-3 

30 

— 


! 

1 


226-2 

264-8 

303-9 

[' 







































Weight of Flat Rolled Iron per Foot. 


175 


cn 


0 

0 

fe 

u 

© 

& 

a 

o 

* 

rt 

'S 

© 

pH 

© 

IS 

d 

s 

0 


© 

rCj 


i 

a? 

53 


c3 

£ 


r—■* 

<2 

n3 
F4 

«» fl 
« ° 
s.a 

° •S 

• r—« H 

_Cj c$ 

+2 © 

© rQ 

fl 

h ^ .4 

<2 gS 

<S £ P, 

^ fl M 
O G 'd 

3 l§ 

o o 

O P4 


CO 

r—i 

o 

05 

H< 

OO 

co 

b— 

cq 

CO 

Hi 

CO 

(N 

co 

o 

Hji 

CO 

CN 

Hi 

— 

CO 

<o> 

O 

05 

CO 

00 

b- 

b- 

CO 

CO 

tO 

O 


05 

co 

CO 

CN 

CO 

rH 

CO 

to 

o 

05 

b- 

CO 

to 

Hi 

co 

CN 

• 

rH 

o 

to 

• 

• 

co 

• 

co 

• 

<N 

• 

CN 

• 

r-H 

• 

pH 

to 

• 

o 

CN 

r—i 

rH 

rH 

rH 

rH 

rH 

rH 

rH 


05 

CO 

b~ 

CO 

to 

Hi 

CO 

CN 

rH 

H 


CO 

tO 

tO 

tO 

Hi 

Hi 

XT' 

CO 

co 

CO 

to 

CN 

05 

CO 

CO 

05 

CO 

H 

• 


o 

CO 

o 

O 

O 

O 

o 

O 

o 

oq 

<N 

CN 

pH 

rH 

rH 

o 

o 

O 

CO 

cn 

oo 

b*. 

CO 

to 

Hi 

CO 

cq 

r-H 

o 

o 

o 

• 

O 

o 

• 

05 

O 

o 

cp 

tO 

• 

o 


-rH 

t-H 

rH 

rH 

rH 

pH 

r— 1 

j-H 

rH 

05 

CO 

b- 

CO 

tb 

Hi 

cb 

CN 

rH 

pH 



co 

rH 

CO 

rH 

CO 

r~; 

tO 

tO 

Hi 


CO 

CO 

CN 

CN 

T—1 

o 

»o 

CO 


■*HH t-H 

CN 

CN 

CO 

CO 

Hi 

Hi 

o 

to 

O 

to 

o 

tO 

O 

to 

CO 

CN 

05 


CN 

% 

co 

tO 

H 

co 

CN 

• 

rH 

• 

o 

to 

• 

uO 

• 

CO 

• 

co 

• 

b- 

• 

• 

CO 

• 

CO 

05 

Hi 

• 

CO 



rH 

T*H 

r 

rH 

r- 

pH 

rH 

05 

00 

b» 

CO 

to 

Hi 

CO 

CN 

T —i 

pH 

o 




co 

CO 

b- 

b- 

1>* 

Hi 

b- 

05 

rH 

co 

CO 

co 

rH 

CO 

to 

CO 

o 



HOC 

co 

Hi 

*o 

CO 

b- 

b^ 

b- 

b- 

CO 

CO 

CO 

co 

05 

05 

05 

Hi 

to 



CN 

CN 

CO 

cq 

PH 

• 

o 

OO 

• 

05 

o 

• 

rH 

• 


CO 

• 

Hi 

• 

to 

CO 

i>- 

co 

• 

05 




rH 

rH 

rH 

rH 

pH 

05 

cb 

CO 

•b— 

CO 

to 

Hi 

do 

c?q 

H 

rH 

o 





1> 

CO 

CO 

Hi 

Hi 

oo 

Hi 

CO 

HH 

05 

Hi 

05 

Hi 

05 

b^ 

to 




CN 

CO 

CO 

05 

rH 

05 

Hi 

o 

to 

r 

co 

CN 

b- 

co 

OO 

CO 

Hi 





CN 

• 

rH 

• 

O 

• 

O 

CN 

• 

H* 

• 

CO 

• 

b^ 

• 

05 

• 

o 

• 

CN 

• 

co 

• 

to 

• 

CO 

CN 

op 





rH 

rH 

rH 

rH 

05 

CO 

b- 

CO 

tO 

to 

H* 

co 

CN 

r -1 

rH 

C5 






05 

O 

tO 

CO 

rH 

05 

CO 


<N 

o 

CO 

co 

Hi 

CO 






H* 

O 

CO 

O 

rH 

CN 

(N 

co 


to 

co 

CO 

b- 

CO 

OO 

05 





r-H 

• 

rH 

O 

tO 

• 

05 

• 

rH 

• 

CO 

• 

to 

• 

b- 

05 

r-H 

• 

co 

to 

• 

rH 

b- 

• 






rH 

rH 

05 

GO 

b- 

b- 

CO 

to 

hH 

cb 

co 

dq 

p—1 

pH 

o 







O 

rH 

cn 

CO 

co 


oo 

to 

CO 

b^ 

CO 

05 

05 

05 






CCft< 

rH 

b- 

CO 

05 

to 

rH 

b- 

CO 

05 

to 

r 

b^ 

O 

CO 






rH 

vO 

OO 

rH 

CO 

CO 

05 

rH 


CO 

05 

CN 

Hi 

pH 

b- 







05 

do 

oo 

b- 

cb 

to 

tb 


cb 

CN 

CN 

rH 

rH 

o 








b- 

rH 

Hi 

CO 

<N 

>o 

05 

CN 

CO 

05 

CN 

05 

CO 







tO{QO 

CO 

tO 

o 

b^ 

05 

o 

rH 

co 

Hi 

tO 

b— 

CN 

CO 

<4H 






rH 

cn 

tO 

00 

rH 


co 

rH 

Hi 

b- 

O 

CO 

O 

CO 

o 







00 

b~ 

cb 

cb 

tb 


HH 

CO 

CN 

CN 

rH 

rH 

o 

fH 

• 







O 

b- 

CO 

05 

to 

CN 

CO 

to 

O 

O 

05 

co 

© 







I-((N 

b- 

CO 

o 

CO 

-co 

O 

CO 

CO 

O 

CO 

Hi 

CO 

rO 







'.*-H 

05 

co 

b- 

O 

o 

• 

co 

• 

rH 

• 

to 

• 

05 

• 

CN 

05 

CO 

• 

a 








CO 

cb 

to 

tb 


co 

co 

CN 

1— 

rH 

o 

o 

H 









co 

b— 

CO 

to 

Hi 

Hi 

to 

CN 

PH 

o 

o 









CO** 

o 

(N 


co 

OO 

O 

CN 

Hi 

CO 

b- 

CO 

W 








rH 

co 

(N 

CO 

o 

Hi 

05 

CO 

b- 

rH 

CO 

to 










• 

« 

* 

• 


• 

« 

• 

• 

• 

• 










to 

tO 



co 

CN 

CN 

rH 

rH 

o 

o 











Cs 

“t 

CO 

co 

O 

CN 

Hi 

CO 

CN 

CO 

'Hi 










to 

(N 

05 

CO 

Hi 

rH 

CO 

tO 

05 

CN 










rH 

b- 

CN 

CO 

rH 

CO 

pH 

to 

• 

o 

• 

b- 

• 

to 

• 

is 










4fi 

HH 

cb 

CO 

CN 

CN 

rH 

rH 

o 

o 












CN 

o 

o 

tO 

rH 

tO 

O 

CN 

to 











HK 

O 

CN 

to 

b- 

O 

CN 

tO 

rH 

b- 











CO 

CO 

CO 

CO 

-05 

Hi 

05 

b* 

Hi 

a< 

o 










t-h 

cb 

CO 

<N 

CN 

• 

rH 

• 

rH 

- O 

•O 

o 

HH 












CO 

Hi 

CN 

o 

b- 

•O 

co 

CN 













to 

CO 

rH 

05 

co 

Hi 

CO 

CN 

© 











rH 

05 

to 

rH 

CO 

CN 

CO 

CO 

Hi 

5 












cq 

CN 

Hi 

rH 

pH 

O 

o 

o 


CN 

* 

CN 


CO b— CO 

Hi b- O 
CO Hi T—I 


CO 

CO 


CO 05 
uo CO 
to CO 

o o o 


«N* 


•H *o o 
CO CO tO 
o cn 05 


co 

CO 

CO 


H* CO 

1> rH 

Hi co 


rH O O O O 


CO CN 
to 05 
O b- 


00 

cn 

to 


CO H< 

05 CO 

co cn 


o o o o 


CO 

<0 


CN 

cn 

Hi 


o o 


CO rH 
rH rH 
CO CN 
• • 
o o 


acts co 
o 


b- 00 
CO to 

CN rH 
• • 

o o 


CN 00 
CO o 

pH pH 
• • 

o o 


00 

O 




15 * 







































Weight of Plat Rolled Iron per Foot, 

Rule.—L ook first for the thickness, and follow the line 
until the column where the hreadth is on the top, is the num- 


17-6 


Weight or Flat Rolled Iron per Foot. 


rH r-lOOCOrHCPrHQOCOCOOOOtOCO® 
rHrHrHrHO5G5GOQ0iH in Xh i>» ZO CO CO ZO 


CO Xh 

QO tb 
tO tO 


CO oa 
pq CO 

CO o 
tO to 


to 


05 

o 

CO 

o 

rH 

00 

CO 

o 

b- 

rH 

rH 

00 

to 

pq 

cb 

pq 

rH 

• 

co 

• 

rH 

• 

to 

• 

X- 

• 

CO 

• 

GO 

rH 

• 

O 

• 

to 

• 

rH 

b- 

• 

o 

o 

X- 

pq 


pq 

X- 

to 

pq 

O 

00 

to 

cb 

o 


rH 

05 

05 

oo 

00 

X- 

b- 

b- 

b^ 

co 

CO 

CO 

CO 


o 

CO 

CO 


X- rH 
GO rH 

to CO 
to to 


iH CO 

O to 

rH CO 

O rH 


C0C005rH05iOPq®iHtOC0rHG0C0C0rH05iH 

Hrqt005Pqc005C0®iHC0®i>.rHOiHrHrHiHrH 

■•'lHPqGOCOCOrHPq05iHtOCq®GOtOCOrHGOCO 
0505G0C0iHiHiHC0CCC0C0CCtOtOtOtOrH rH 


o 


rH 1 h rH 
N W OD 

cb rH 05 
go go in 


o in 

rH as 

• • 

to o 
iH in 


to CO 
Ih to 

QO CO 
ZO CO 


H O 
CO r-i 
• • 
rH Pq 
co co 


GO CO 

co co 
• -* 
05 b- 

to to 


rH CO 
rH pq 

tb cb 

tO to 


rH 05 

o b- 
• 

rH GO 
tO rH 


iH to 

to co 
• • 
CO rH 
rH rH 


to 


CO rH 

Pq O 
• • 

O co 

GO in 


rH 05 
GO to 

-• a 

rH 1 > 

in co 


lH CO 
05 CO 

tb cb 
co co 


tO rH 

<M rH 
• • 

rH 05 

CO tO 


CO <M 

O 05 
• • 

iH Hfi 

to to 


O 05 

co CO 

• .• 

pq o 
to to 


CO iH 

to rH 

cb cb 
rH rH 


to 'Tfi 

CO Pq 
,• • 
rH Pq 
rH tH 


co 

*** ^ 
rH Pq 
•Ih 


CN rH 

PJ pq 

bo 4 h 
co co 


o o 
pq pq 

• p 

pq o 

CO CO 


05 GO iH in CO CO tO tH CO CO 


co co 
to to 


rH pq 
-tO »o 


O CO co 
to rH *rH 


rH pq 

rH rH 


o 

rH 


rH 


co CO 

co oo 

• *• 

rH ® 
CO co 


CO CO 
05 O 

CO in 

to to 


co co 

rH cq 

tb cb 
to to 


rH to 

oq oq 

• • 

rH 05 

to rH 


co pq pq 
rH co in 

b- tb co 

H H ^ 


Pq pq 

GO 05 

• • 

rH 05 
tH CO 


pq 

o 

CO 

co 


to 

HH ^ 
rH i>* 
tO 


to cp 

co 00 
• * 

to CO 

to to 


CO X'- 

o pq 
• • 

pq o 
to to 


J>. GO 
rH CO 

oo cb 
rH rH 


GO 05 05 

go ® pq 
• * * 
rH CO rH 
rH rH rH 


o o 
to 


b- 05 


05 i>* 

co co 


to 

CO 




00 

05 

o 

rH 

pq 

CO 

rH 

tO 

CO 

iH 

co 

05 

CO 

CO 

o 

co 

CO 

05 

pq 

tp 

oo 

rH 

rH 

Xh 

pq 

o 

05 

b- 

tb 

CO 

pq 

o 

cb 

iH 

tb 

cb 

to 

to 

rH 


rH 

rH 

rH 

rH 

CO 

co 

co 

co 


05 

Hqo 

CO 05 
HH 


co co 

rH GO 
• • 

i>- to 
rH hH 


05 rH 
rH to 
• • 
rH pq 
rH hH 


pq co hh 
05 oq co 
• • • 
O 05 b- 
rH CO CO 


O CO 
• • 

CO rH 
co co 


CO 

b^ 

Pq 

CO 


C0W» 

CO 


CO to 
05 CO 

»b rH 

rH rH 


b- 05 
b- rH 
.• • 
Pq rH 

rH rH 


® pq oo 

CO O rH 

05 cb o 
CO CO co 


to co 
co pq 

hH cb 
co co 


co 

CO 

• 

rH 

CO 


'1 v I- t' 

A * •* *' 


N 

*#» « 
*o pq 
rH 


hH rH 

CO CO 

# • 

rH 05 
hH CO 


CO tO rH 

pq b- pq 

cb <b tb 

co co co 


05 to 
CO rH 

cb pq 

CO CO 




~o 


pq hh 

05 hH 

65 cb 
co co 


co oo o 

05 rH O 
• • • 
CO to rH 
CO CO CO 


CO rH 
tO O 

pq h 
co co 


pq 

co 

o 

co 

b- 

to 

05 

pq 


MfCC 

.CO 


iH 

rH 

rH 

co 

CO 

co 


o 

co 

pq 

iH 

CO 

05 

to 

iH 

tb 

rH 

c<* 

H 

05 

cb 

CO 

CO 

CO 

CO 

co 

pq 

pq 


•CO 


c2 

a3 

Ph 

CO 

0 

O 

Ph 


u 

M 


co 

tO 

iH 

o 

co 

CC 

CO 

05 

to 

pq 

GO 

rH 

rH 

pq 

rH 

o 

GO 

Jh 

CO 

co 

co 

co 

pq 

pq 


rH 

05 

rH 

pq 

O 

/H*> 

05 

• 

rH 

O 

iH 

rr 

• 

CO 

rH 

O 

05 

iH 

CC 


CO 

CO 

pq 

pq 

pq 



pq 

CO 

r—' 

rH 



rH 

OO 

CO 

CO 


.CO 

05 

• 

iH 

• 

CO 

• 

tO 



pq 

Pq 

pq 

pq 




pq 

o 

05 



CSC 

b- 

to 

• 

pq 

• 



pq 

cb 

to 

«H 




pq 

pq 

pq 





05 

co 





co 

• 

pq 

•- 




pq 

rH 

co 





pq 

pq 


*d(X} 

pq 


oo 


pq 

pq 


pq 























Weight of Flat Rolled Iron 


0 

e 

fc 

u 

o 

a 


P 

c3 


g 

3 

*o 

o 

4-3 

Ui 

cd 

0 > 

rp 


C/3 

• r-H 

CQ 

c/3 

<D 

o 

*p 


a> 

H 


Weight of Flat Rolled Iron per foot. 


177 




co 

CM 

05 

O 

rH 

05 

SO 

CM 

® 

tO 

rH 

b- 

rH 

® 

b^ 

co 

rH 

05 

rH 

to 



ih 

CO 

co 

to 

® 

rH 

co 

rH 

05 

co 

co 

Cl 


CM 

SO 

CM 

05 

SO 

® 

co 


co 

lH 

b 

Cl 

05 

cb 

SO 

CM 

© 

t- 

tb 

CM 

® 

lb 

tb 

CM 

® 

tO 

® 

CO 

to 



rH 

rH 

rH 

CO 

CO 

co 

co 

CO 

CM 

Cl 

Cl 

.Cl 

r—i 

rH 

rH 

rH 

ib 

tb 

cb 

CM 



tO 

CM 

05 

co 

co 

H 

co 

to 

Cl 

05 

SO 

co 

® 

b- 

rH 

so 

1 -- 

CO 

co 

05 


cfW • 
to to 

co 

cb 

© 

rH 

co 

cb 

rH 

sb 

© 

rH 

to 

• 

r-H 

t-h 

05 

1 H 

sb 

CM 

tH 

oo 

rH 

rH 

05 

® 

b 

to 

• 

ri 

Cl 

tH 

b- 

CO 

CM 

to 

CO 

rH 

SO 

CM 

rH 



rH 

rH 

rH 

co- 

co 

CO 

CO 

CM 

CM 

CM 

Cl 

rH 

r*-i 

rH 

H 

05 

ib 

rH 

CO 

CM 



rH 

CM 

05 

in 

to 

CO 

© 

CO 

tO 

-CO 

pH 

05 

SO 

rH 

r—i 

rH 

® 

1 H 

to 

co 



CO 

rH 

rH 

CO 

to 

CM 

CO 

to 

ci 

05 

to 

CM 

05 

SO 

05 

lH 

rH 

CO 

CM 


to 

rH 

rH 

05 

lb 

b 

CM 

® 

lb 

tb 

cb 

® 

cb 

sb 

cb 

H 

CM 

05 

SO 

rH 

CO 



rH 

rH 

co 

CO 

CO 

CO 

co 

CM 

Cl 

Cl 

CM 

rH 

H 

rH 

H 

05 

sb 


cb 

CM 



CO 

CM 

® 

CO 

CO 

to 

CO 

rH 

05 

co 

SO 

rH 

cr 

*H 

05 

^H 

co 

SO 

1 h 

. CO 


tO CM 

05 

05 

tH 

t 

iH 

rH 

tb 

CM 

cb 

® 

o 

rH 

00 

cb 

SO 

SO 

co 

rH 

CM 

05 

• 

05 

• 

b^ 

tO 

tb 

CO 

cb 

® 

rH 

tH 

CO 

tO 

SO 

CO 

rH 

CM 

cp 

rH 

CM 



rH 

CO 

co 

co 

co 

co 

CM 

Cl 

CM 

CM 

r —[ 

rH 

rH 

tH 

H 

cb 

sb 

4h 

cb 

Cl 

6 


CO 

CM 

® 

05 

CO 

i>- 

SO 

to 

CO 

CM 

rH 

® 

CO 

1 >- 

SO 

05 

so 

rH 

co 

Cl 

£ 

tO 

rH 

® 

05 

Ih 

CO 

to 

rH 

CO 

CM 

rH 

® 

05 

b— 

SO 

tO 

rH 

GO 

CM 

SO 

r-H 

• r-j 

© 

00 

tb 

cb 

rH 

05 

lb 

tb 

cb 

rH 

05 

sb 


CM 

® 

rH 

co 

CM 

rH 

- 7 " 

Pi 

/-N 


rH 

co 

co 

co 

CO 

CM 

Cl 

Cl 

CM 

CM 

rH 

rH 

rH 

H 

rH 

ci) 

SO 

4h 

cb 



CM 

CM 

rH 

® 

05 

05 

CO 

CO 

b4-. 

. b- 

SO 

tO 

rH 

rH 

co 

SO 

05 

co 

05 

sc # 



H rH 

rH 

rH 

rH 

© 

© 

® 

® 

® 

® 

® 

® 

® 

® 

® 

CM 

H 

rH 

® 

® 


^H 

1 CO 

cb 

rH 

Cl 

© 

cb 

SO 

b 

CM 

® 

cb 

sb 

4h 

CM 

• 

® 

® 

® 

® 

® 

® 

o 

H 


co 

CO 

CO 

CO 

co 

CM 

Cl 

Cl 

CM 

CM 

rH 

H 

rH 

r-H 

H 

cb 

sb 

4h 

-cb 

Cl 

r-~H 

4-5 


Cl 

CM 

rH 

rH 

tH 

rH 

rH 

rH 

H 

rH 

H 

H 

rH 

rH 

1 H 

rH 

co 

CM 

rH 

rH 



CM 

CO 

rH 

tO 

CO 

lH 

00 

05 

® 

rH 

CM 

co 

rH 

® 

O 

® 

® 

*o 

® 

S3 

rtf cb 

rH 

CM 

© 

cb 

cb 

b 

CM 

® 

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05 

• 

b- 

tb 

cb 

• 

rH 

to 

© 

SO 

1 H 

CO 

CO 

© 

05 



CO 

CO 

CO 

co 

Cl 

ci 

CM 

CM 

CM 

rH 

rH 

H 

rH 

rH 

05 

lb 

tb 

CO 

CM 

r-r 

rP 


rH 

CM 

CM 

CM 

Cl 

co 

co 

rH 

rH 

tO 

tO 

SO 

SO 

ir 

to 

H 

so 

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CO 

to 



CO 

tCO 

iH 

05 

tH 

CO 

tO 

1 H 

05 

rH 

co 

to 

1 — 

1 H 

CO 

co 

05 

05 

■ 05 

'“3 

rH rH 

ci 

o 

cb 

• 

CO 

tb 

cb 

• 

rH 

05 

lb 

SO 

'b 

Cl 

• 

® 

05 

• 

H 

• 

cp 

to 

so 

Ip 

a> 


CO 

co 

CO 

CM 

CM 

CM 

CM 

CM 

H 

rH 

r -4 

rH 

rH 

rH 

co 

Jh 

tb 

cb 

CM 

H 



® 

rH 

CM 

rH 

tO 

SO 

lH 

00 

05 

® 

H 

CM 

CO 

rH 

tO 

05 

05 

® 

to 

o 

rO 


rH 

rH 

in 

® 

co 

CO 

05 

Cl 

to 

05 

Cl 

tO 

co 

rH 

rH 

>o 

so 

CO 

co 

05 

o 


® 

cb 

lb 

b> 

cb 

• 

rH 

® 

cb 

SO 

to 

cb 

• 

T“< 

® 

rH 

• 

1 H 

• 

® 

CO 

to 

so 



co 

CO 

CM 

CM 

Cl 

CM 

CM 

CM 

rH 

rH 

rH 

tH 

H 

tH 

co 

SO 

tb 

cb 

CM 

rH 


h< 
t*) V 
CO o 

CO 


so 

rH 

Oi 

CM 


05 


to rH 
• © 

in so 

CM Cl 


to cm 

tO 05 

b CM 
CM CM 


co rH ® 

CM CO O 


iH 

CO 


Hi 

iH 


05 CO CM 
O rH CM 

CO 


rH 05 00 SO H CO rH 


CO iH 
CO rH 
r—< tO 


CMrHrHrHrHrHrHOSCOSO 


O rH 
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O CM 

b CO 


uO iH 

tO CO 

rH CO 

• • 

CM rH 


05 

*** 

CO 05 
CM 


tO 

OO 

Cl 


CO tO 

H< co 
cb tb 

CM CM 


so co 

iH H 

cb ci 

CM CM 


05 

tO 

o 

CM 


rH CM 
O 

05 ib 


Hi 

co 


iH 

CM 


tO rH 


SO 

CM 


05 tO 

o o 

rH 05 


rH CO 
Cl CO 

05 CO 

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in so 


ci co 
to co 


O rH 

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CO to 


rH CO CM 


rH 

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co co 

Cl 


CO 

to 

ib 

CJ 


CO o 
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to rH 
CM CM 


iH 

05 


rH 

rH 


O iH rH 

05 co co 


CM 
CM CM 


rH 05 CO CO 


CO 

to 


05 

-tH 

cb 


o 

CM 

CM 

rH 


CM 00 
in GO 


iH to 

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CO rH 


rH CM 
05 CO 

to o 


1- rH 
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CM to 


rH05l>*COrHCOCliH 


05 

Hm <? 
CO co 
CM 


CO 

cb 

CM 


CO to 
rH CO 

tO CO 
CM CM 


CO o 

rH 1 h 

Cl o 

CM CM 


CM rH CO 
CM ip CM 

05 -tb SO 


co 

ip 

rH 


op 

cb 


CO tO rH 

CO CO N 
• - * 00 


CO rH 
05 rH 
CO 05 


CO in 
CO to 
rH 05 


CO 05 

rH 1 h 
CM rH 


CO to rH CM CM r—i 


CO 

«*» V 

CO iH 
CM 


CO 

CO 

ib 

CM 


CO rH 
CM CO 

rH CM 
CM CM 


CO co 

CO 05 


CO 

to 


o co 

rH CO 


»o 

CM 


co 

co 


i—IC5COb-tOrHCMH 


® O rH 
rH CO to 
05 to 


05 CO 
CM O 


N r —1 

in to 

CM CO 


CM rH rH rH rH rH rH rH 05 CO 1 h *0 rH Cl 


CO CO 
CO Cl 
H rH 

CM H 


CO 

HH ? 
CO 

CM 




rH 

CM 


rH 1h 
CO 05 

cb rH 
CM CM 


05 CM 

tO CM 


tO in O 
00 rH H 


CO 

b- 


® 

Cl 


05 N CO tO CO 


SO 

co 

CM 

.rH 


CO ® iH 
05 rH CO 

© *P 

rH 05 C/0 


tO CM 
CO 05 
CO rH 


05 CO 
Jfri rH 

rH iH 


05 CO 
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CO to rH CM Cl rH 


IH 

so 

rH 

Cl 

® 

CO 

SO 

rH 

® 

1 ^ 

rH 

rH 

co 

rH 

rH 

CO 

tb 

CO 

CM 

H 

05 

cb 

lb 

tb 

CM 

Cl 

CM 

CM 

rH 

rH 

rH 

H 


® 

CM 


CO 

00 


CO rH ® 


tO ® ® 

tO rH CM 
iH 05 


rH rH 05 1h 


rH 


rH 


. rH ® 
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CO Cl 

cb tb 

tb 


® ® 
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05 CC 
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® ® 
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05 CM to 


CO 

® 

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Cl 

CO 

Cl 

Cl 

21-5 

CM 

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® 

05 

rH 

ip 

b^- 

rH 

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sb 

r — 1 

CM 

tb 

r-H 

05 

tb 

H 

ZO 

Cl 

rH 

rH 

• 

tH 

H 

tH 

® 

H 

in 

CO 

do 

® 

co 

b 

CO 

CO 

sb 

so 

® 

tb 

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cb 

CO 

tO 

CM 

® 

05 

H 

SO 

CM 

H 


b- 

so 

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co 

CM 

® 

05 

b— 

so 

rH 

CO 

so 

CM 

tH 

CM 

co 

Hi 

05 

CM 

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NX 

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CO 

so 

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H 

05 

H 

® 

CO 

IH 

to 

rH 

CM 

Cl 

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• 

• 

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• 

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co 

so 

Hi 

CO 

CM 

CM 

cb 


® 

05 

CO 

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CO 

CM 

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> 


• 








Cl 

Cl 

CM 

rH 

H 

H 

H 

H 

r—' 

rH 

H 

05 

do 

IH 

sb 

rH 

cb 

CM 

r—< 

—H 




>o 

CO 

CM 

SO 

® 

rH 

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CM 

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Cl 

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CO 

IH 

tO 

® 

r—r 

Cl 

rcH* 

o 

05 


tO 

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CM 

H 

05 

b- 

SO 

rH 

05 

co 

tH 

® 

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CO 

CM 

Hi 

so 






© 


• 

• 

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v 

• 

Cl 

r-H 

05 

co 

SO 

rH 

CO 

1 ^ 

r — 1 


Cl 

® 

05 

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b- 

SO 

tO 

CO 

CM 

H 

® 

• 

©• 



• 

• 


• 

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CM 

CM 

H 

T— 1 

tH 

rH 

H 

rH • 

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H 

r—J 

05 

CO 

sb 

MO 

rH 

CO 

CM 

rH 

rH 


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CO 

Cl 

CM 

r — J 


05 

H 

Cl 

co 

rH 

to 

O 

CM 

cc 

CC 

05 


® 

05 

CO 

1h 

SO 

to 

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CO 

Cl 

® 

CO 

1— 

so 

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co 

IH 

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SO 

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• 

• 



© 

• 

• 

• 

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05 

co 

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CO 

CM 

SO 

H 

Cl 


05 

CO 

lb 

so 

to 

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co 

Cl 

rH 

• 


• 

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H 

rH 

H 

H 

tH 

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rH 

H 

rH 

05 

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b— 

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to 

b 

CO 

CM 


r-H 


SO 

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05 

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CO 

co 

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CM 

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CO 

Cl 

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CC 

CM 

Hi 

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05 

CO 

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lH 

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co 

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r-H 

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CO 

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Cl 

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05 


sb 

tb 

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cb 

CM 

r-H 

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• 

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Cl 

rH 

tH 

tH 

rH 

rH 

pH 

tH 

rH 

rH 

05 

CO 

tH 

so 

to 

b 

CO 

CM 

— 1 

*H 




Hx> 


HD 

W 

ifijCO 

HN 

ccja: 


Hx 


Ha 


tdfXi 

He* 

?£!9 

Hf 


Ha 


Cl 

CM 

CM 

CM 

rH 

rH 

?H 

rH 

rH 

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iH 

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178 


Weight of mateeiues. 


Weight Per Square Foot in Pounds. 


Thickneua 
in inches. 

Cast Iron. 

Wrought or 
Sheet Iron. 

Sheet Copper. 

Sheet Lead. 

Sheet Zinc 

1 

1 6 

2-346 

2-517 

2-890 

3-694 

2-320 

4 

4-693 

5-035 

5-781 

7-382 

4-642 

3 

Iff 

7-039 

7-552 

8-672 

11-074 

6-961 

i 

9-386 

10 070 

11-562 

14-765 

9-275 

J 

* T6 

11-733 

12-588 

14-453 

18-456 

1L61 

t 

14-079 

15-106 

17-344 

22-148 

13-93 

T 

TB 

16-426 

17-623 ; 

20-234 

25-839 

16-23 

4 

18-773 

20 141 

23-125 

2y*os0 

18-55 

V 

T?r 

21-119 ^ 

22-659 

26-016 

33-222 

20-87 

f 

23-466 

25-176 ■ 

28-906 

36-913 

23-19 

u 

25-812 

27-694 

31-797 

40-604 

25-53 

f 

28-159 i 

30-211 

34-688 

44-296 

27-85 

13 

15 

i 

30-505 

32-729 

37-578 

47-987 

30-17 

32-852 

35-247 

40*469 

51-678 

32-47 

fl 

35-199 

37-764 

43-359 

55-370 

34-81 

1 

37-545 

40-282 

46-250 

59-061 

37-13 


42-238 

45-317 

52-031 

66-444 

41-78 

14 

46-931 

50-352 

57*813 

73-826 

46-42 

if 

51-625 

55-387 

63-594 

63-594 

51-04 


56-317 

60-422 

69-375 

88-592 

55-48 

it 

61-011 

65-458 

75-156 

95-975 

60-35 

if 

65-704 

70-493 

80-938 

103-358 

65.00 

if ; 

70-397 

75-528 

86-719 

110-740 

69-61 

2 

75-090 

80-563 

92-500 

118-128 

74-25 


Weight of Copper Rods or Bolts per Foot* 


Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter 

Weight. 

Inches. 

Pounds. 

Inches 

Pounds. 

Inches. 

Pounds. 

Inches. 

Pounds. 

i 

0-1892 

l 

3-0270 

if 

10-642 

3f 

34-487 

~E 

TS 

0-2956 


3-4170 . 

2 

12-108 

3i 

37-081 

f 

0-4256 

li : 

3-8912 : 

2i : 

13-668 . 

3f 

39-737 

7 

15 

0-5794 

lf 6 

4-2688 

2i 

15-325 

3f 

42.568 

i 

0-7567 

li 

4-7298 

2f 

17-075 

3f 

45'4.55 

9 

rs 

0-9578 

1 * 

l T6 

5-2140 

2i 

18-916 

4 

48-433 

f 

1-1824 

if 

5-7228 

2f 

20-856 . 

4i ’ 

53-550 

i! i 

1-4307 

A T6 ; 

6-2547 

2f i 

22-891 

44 

61-321 

f 

1-7027 

14 

6-8109 ; 

2£ 

25-019 

4f 

68-312 

1 J 

1 5 

1-9982 

i t 9 6 

7-3898 

3 

27-243 

5 

76*130 

i 

2-3176 

if 

7-9931 

3f 

29-559 

54 

91-550 

1 5 

16 

2*6605 

u 

9-2702 

3i 

31-972 

6 

109- 



































Birmingham Gauge for Wire, Sheet Iron and Steel* 179 



Weight per Square Foot in Founds. 


Thickness by 

Thickest in 

Sheet and 

Sheet Catt 

Sheet 

Thickne 

Ihe Gatye. 

Inches, 

Boiler Iron. 

Steel. 

Copper. 

in lnche 

Ao* 0 

0*340 

13-7 

14-0 

i 15-6 

H 

“ 1 

0*300 

12-1 

12-4 

13-8 

j 

l 6 

“ 2 

0.284 

11-4 

, 11-7 

130 

9 

3 

0-259 

10.4 

10-6 

11-9 

i 

“ 4 

0-238 

9-60 

9 80 

11-0 

T 

“ 5 

0 - 22(1 

8-85 

9-02 

101 

66 

“ 6 

0-203 

8-17 

8-33 

9-32 

66 

“ 7 

0-180 

7-24 

7-38 

8-25 

3 

T9 

** 8 

0*165 

605 

6-78 

7-59 

•4 

“ 9 

0-148 

5-96 

6-08 

6*80 

s 

“10 

0-134 

5-40 

5-51 

6-16 

s z 

u 

“11 

0-120 

4-83 

4-93 

5-51 


“ 12 

0-109 

4-40 

4-50 

5-02 

u 

“13 

0-095 

Q.«q 

o.qi 


z 




O irl 

4-37 


“14 

0-083 

3-34 

3-41 

3-81 

46 

“15 

0-072 

2-90 

2-96 

> 3-31 

1 

“16 

0-0-65 

2 - 6-2 

2-67 

; 3-00 

66 

“17 

0-058 

2-34 

2-39 

2-67 

66 

“18 

0-049 

1-97 

201 

2-25 

66 

“19 

0-042 

1-69 

1-72 

1:93 

z 

“20 

0-035 

1-41 

1-42 

1-61 

<5 ¥ 
a 

“21 

0-032 

1-29 

1-31 

1-47 

44 

“22 

0-028 

1-13 

115 

1-29 

X 

“23 

0-025 

1-00 

102 

1-14 

«> 4 

66 

“24 

0-022 

0-885 

0-903 

1-01 

44 

“25 

0-020 

0-805 

0-820 

0-918 

66 

“26 

0-018 

• 0-724 

0-738 

0-820 

61 

“27 

0-016. 

0-644 

i 0-657 

0-735 

46 

“28 

04)14 

0-563 

0-574 

0-642 


“ 2 9 

6 Q Cl 

0-013 

0-523 

0-533 

0-597 




0-483 

0-493 

0-551 


31 

0-010 

0-402 

0-410 

0-480 


32 

0-009 

0-362 

0-370 

0-42.0 


“33 

0-0 Q 8 

0-322' 

0-328 

9-370 


“34 

0-007 

0-282 

0-288 

0-323 


“35 

0-005 

0-230 

0-235 

0*262 


“36 

0-004 

0-170 

0-173 

0-194 






















Proportion of Bolts and Nats. Number of threads per In# 





















































Table for Falling Bodies. 


181 


Velo¬ 

city 

Space fall- 

Time in 

Velrcity 

Space fall- 

Time in 

Velocity 

Space fall. 

Time in 

»t the 

end. 

en through 

sec >nds 

at the 
end. 

en through 

second*. 

at the 
end. 

en through 

seconds. 

V 

s 

T 

V 

S 

T 

V 

S 

T 

0-1 

•00015 

0-0031 

5-1 

•40388 

0-158 

11 

1-8789 

0-342 

0.2 

•00062 

0-0062 

5-2 

•41987 

0-162 

12 

2-0652 

0-373 

0-3 

•00139 

0-0093 

5-3 

-43618 

0-165 

13 

2-6242 

0-405 

0-4 

•00248 

0-0124 

5-4 

•45279 

0-168 

14 

3-0435 

0-436 

0'5 

•00388 

0-0155 

5-5 

-46972 

0-171 

15 

3-4938 

0-467 

0-6 

•00559 

0-0187 

5-6 

•48695 

0-174 

16 

3-9751 

0-498 

0-7 

•00761 

0-0218 

5-7 

•50450 

0-177 

17 

4"4876 

0-530 

0-8 

•00994 

0-0230 

5-8 

•52236 

0181 

18 

5-0310 

0-560 

0-9 

•01257 

0-0280 

5-9 

•55057 

0-184 

19 

5-6056 

0-591 

l- 

•01552 

0-0311 

6- 

•55900 

0-187 

20 

6-2112 

0-622 

M 

•01879 

0-0342 

6-1 

•57779 

0-190 

21 

6-8478 

0-654 

1-2 

•02065 

0-0373 

6-2 

•59689 

0-193 

22 

7-5155 

0-685 

1-3 

•02624 

0-0404 

6-3 

•61630 

0-196 

23 

8-2143 

0-716 

1-4 

•03043 

0-0436 

6.4 

•63602 

0-199 

24 

8-9441 

0-747 

1-5 

•03493 

0-0467 

6-5 

-65606 

0-202 

25 

9-7049 

0*778 

J-6 

•03975 

0-05 

6-6 

•67639 

0-205 

26 

10-497 

0-810 

1-7 

•04487 

0-052 

6:7 

•69705 

0-209 

27 

11-320 

! 0-840 

1-8 

•05031 

0-556 

6.8 

•71801 

0-212 

28 

12.174 

0-872 

1-9 

•05605 

0-0591 

6-9 

•73928 

0-215 

i 29 

13-059 

0-903 

2- 

•06211 

0-0623 

7- 

•76087 

0-218 

30 

13-975 

j 0*933 

2-1 

•06847 

0’0654 

7.1 

•78276 

0-221 

31 

14-922 

0-965 

2-2 

•07515 

0-0685 

7-2 

•80497 

0-224 

32 

15-900 

0-996 

2*3 

•08214 

0-0717 

7-3 

: -82748 

' 0-227 

33 

16-910 

1-025 

2-4 

•08944 

0-0747 

7-4 

! -85031 

0-231 

34 

18-789 

1-058 

2-5 

•09705 

0-0780 

• 7-5 

; -87344 

0-234 

35 

19-022 

1-091 

2-6 

•10497 

0-0810 

7-6 

•89689 

0-237 

36 

20-124 

1-120 

2-7 

•11320 

0-0841 

7-7 

i -92065 

0-240 

37 

21-258 

1-151 

2-8 

•12174 

0-0872 

7-8 

•94472 

0-243 

38 

22-422 

1-184 

2-9 

•13059 

0-0903 

7-9 

•96910 

0-246 

39 

23-618 

1-213 

3- 

•13975 

0-0934 

8- 

•99379 

0-250 

40 

24-844 

1-243 

3-1 

•14922 

0-0966 

8-1 

1-0187 

0-253 

41 

26-102 

1-276 

3-2 

•15900 

0-0997 

8-2 

1-0441 

0-256 

42 

27-391 

1-308 

3-3 

•16910 

0-1025 

8-3 

1-0697 

0-259 

43 

28-57 

1-338 

3-4 

•18788 

0-1059 

8.4 

1-0956 

0-262 

44 

30-062 

1-370 

3-5 

•19022 

0-1092 

8 5 

1-1218 

0-265 

45 

31-444 

1-400 

3-6 

•20124 

01121 

8.6 

1-1484 

0-268 

46 

32-857 

1-431 

3-7 

•21257 

0-1152 

8-7 

1-1753 

0-271 

47 

34-301 

1-463 

3-8 

•22422 

0-1185 

8-8 

1-2015 

0-274 

48 

35-776 

1-495 

3-9 

•23618 

0-1214 

8-9 

1-2299 

0-278 

49 

3 "-282 

1-525 

4* 

•24844 

0-1246 

9- 

1-2577 

0-281 

50 

3 S -820 

1 -555 

41 

•26102 

0-1278 

9-1 

1-2858 

0-283 

- 51 

40-388 

3-588 

4-3 

•27391 

0-1309 

9-2 

1-3143 

0-287 

52 

41-987 

1-619 

4-3 

•28571 

0-1339 

9-3 

1-3430 

0-290 

53 

43-618 

1-650 

4-4 

•30062 

0-1371 

9-4 

1-3720 

0-293 

54 

45-279 

1 -680 

4-5 

•31444 

0-1403 

9-5 

1-4041 

0-296 

55 

46972 

1-711 

4-6 

•32857 

0-1433 

9-6 

1-4310 

0-300 

56 

48-695 

1-742 

4-7 

•34301 

0-1465 

9-7 

1-4610 

0-302 

57 

50-450 

1-774 

4.8 

•35776 

0*1496 

9-8 

1*4913 

0-306 

58 

52-236 

1-805 

4-9 

•37282 

0-1526 

9-9 

1-5219 

0-309 

59 

55*058 1 

1-835 

5- 

•38820 

J 

0-1559 

_ 

10 

1-5528 

0-312 

60 

55-900 J 

1-868 


























182 


Gravitation. 



GRAVITATION. 


Gravity or Gravitation is a mutual faculty which all bodies in nature 
possess, to attract one another; or Gravity is the force hy which all bodies 
tend to approach each other, A large body attracting a comparatively very 
small one, and their distance apart being inconsiderable, the force of gravity in 
the small body will lie very sensible compared with that ih the large one ; such 
is the case with the body, our earth, attracting small bodies on or near her sur¬ 
face. 

Gravitation is not periodical, it acts continually ever and ever. A body placed 
unsupported at a distance from the earth, th & force of gravity is instantly oper- 
; ating to draw it down, and then we say, “ the body fell down ” If it were possi- 
i ble to withdraw the attraction between the body and the earth, it would not 
j fall down, but'remain unsupported in the space where it was placed;—giving 
I the body a motion upwards it would continue that, and never come back to the 
earth again. 

Law of Gravity* 

The ferret of Gravity is proportional to the mass of the attracting todies, and in¬ 
verse as the square of their distance apart. 

This law was discovered by Sir Isaac NewtOn. It is this law that supports the 
condition of the whole universe, and enables us to calculate the distances, mo¬ 
tions and masses, &c., of the heavenly bodies. 

The unit or measure of force of gravity is assumed to be the velocity a falling 
body has obtained at the end of the first second it falls; this unit is commonly 
denoted by the letter g ; its value at the level of the sea in New York is 
g = 32-166 feet per second, in vacuum. The space failed through in the first 
second is |g = 16‘083 feet. 

This value augments with the latitude, and abates with the elevation above 
the level of the sea. 

I = latitude, h = height in feet above the level of the sea, and r = radius of 
the earth in feet, at the given latitude l. 

r = 20887510(1+0-00164 cos.21), 

0= 32-16954(1 —0-00284 cos.27)(l ——■) 

Letters denote. 


S — the space in feet, which the falling body passes through in the time T. 
u — the space in feet, which the body falls in the STth second. 

V = velocity in feet per second, of the falling body at the end of the time T. 

T = time in seconds the body is falling. 

The accompanying Diagram is a good il¬ 
lustration of the acceleration of a falling 
body. The body is supposed to fall from a 
to b, every small triangle represents the 
Space 16-08 feet which the body falls in 
the first second; when the body has reached 
the line 3" seconds, it will be found that it 
has passed 9 triangles, and 9X16-08 = 144-72 
feet the space which a body will fall in 3" 
seconds. Thenumberof triangles between 
each line is the space u which the body has 
fallen in that second. Between 3" and 4" 
are 7 triangles and 7X16-08 =* 112-56 feet, 
the space fallen through in the fourth sec¬ 
ond. Under the line 3" will be found 6 tri¬ 
angles, which represents the velocity V the 
body has obtained at the end Of the third 
second or 6X16-08 = 90"48 feet per second. 
For every successive second the body will 
gain two triangles or 2X16-08 — 32-16 feet 
pul second. 


u 

r 

S 

T 

1 

2 

1 

1" 

3 

4 

4 

2" 

5 

6 

9 

3" 

7 

8 

16 

4" 

9 

10 

25 

5" 

11 

12 

36 

6" 





























Gravitation. 


183 


Formulas for Accelerated Motion. 

F= g T= ~ = = 8-02 y'S, 


S: 


g T* 

2 


V T 
~2~' 


F» 

2 g 


jra 

64-33 * 


V 

- 

g 


2 S 
V 



u = g(T— £), 


2_S 

u 


rs 

4-01’ 




1, 

2 , 

4. 


Example 1. A body is dropped at a height of 98 feet. What velocity will it 
have when it reaches the ground, and what time will it take to fall down ? 

Formula 1. V= 8*02 == 8-02 yW = 79-39 feet.per second. 


Formula 3. 


T ■= 


_ vs. 


4-01 


y98 

im 


2-46 seconds. 


Example 2. A body was dropped at the opening of a hole in a rock, and 
reached the bottom after 3‘5 seconds. How deep was the hole? 

Formula 2. S = 9 — = —- 16 - * 3 ' 0 ? = 196*98 feet. 

2 2 

Retarded Motion. 

A body thrown up vertically will obtain inversely the same motion as when 
it falls down, because it is the same force that acts upon it, and causes retarded 
motion when it ascends, and accelerated motion when it descends. 

V = the velocity at which the body starts to ascend. 
v == velocity at the end of the line t. 

T — time in seconds in which the body will ascend. 
t — any time less than T. 

S = height in feet to.which the body will ascend, 
s = the space it ascends in the time t. 

Formulas for Retarded Motion. 


v = V- 


, s gt 


q 


t r-f 


g& 


V=v+gt~ f+f, 


t = 


v — v __ v _ /r «_2 s 

9 ~ g l~~g' 


5, 


6, 


7 , 


8 . 


Formulas for T and S, is the same as for accelerated motion. 

Example 3. A ball starts to ascend with a velocity of 135 feet per second. 
At what velocity will it strike an object 60 feet above? Find the tenet,by the 
Formula 8. 


t 


135 __ / 135^ 

32*16 \/ 32 - 16 a 


2X60 


= 0-41 seconds, until it strikes, and from 


Formula 5, we have, 

v = 135 


32-16 

32-16X0-41 = 121-83 feet, per second. 












184 


Gravitation. 


Example 4. A ball thrown up vertically from a cannon, occupied 9 seconds, 
until it arrived at the same place it started from. How high up was the ball, 
and at what velocity did it start? 

One half of 9 = 4^ seconds. Formula 2. 


S = 


32-16X4-53 


326 feet high. 


V = 32-16X4-5 = 144-7 feet per second. 

If a cannon ball be shot from A, in the direction AB, at an angle BA C to the 
horizon, there are two forces acting on the ball at the same time, namely,—the 
force of gunpowder, which would propel the ball uniformly in the direction A B, 
and the force of gravity which only acts to draw the ball down at an accelerated 
motion; these two different (uniform and accelerated) motions will cause the 
ball to move in a cui-ved line, (Parabola) AaC. Fig. 225. 

V = velocity of the ball at A. W— weight of the ball in pounds. 

S = the greatest bight of ball over the horizontal line AC. 

t = time from A to C. via a. p — pounds of powder in the charge, 

b — the distance from A to C, called horizontal range . 

/ V " WV* . p 

\ = 2800 ^/ yy ’ p = ' 7840000 , cos. x jy 

Example 5. The cannon being loaded sufficiently to give the ball a velocity of 
900 feet per second, the angle x — 45°. Required the distance b = ? and the 
time t = ? 

b — - bn'ia -— 4259 feet, the distance from A to C. 

6'Z'lo 

It will be observed that the distance b will be longest when the angle x is 45°, 
because the product of sine and cosine is greatest for that angle. sin.45°Xcos. 
45° = 0-5. 

Example 5. What time will it take for a ball to roll 38 feet on an inclined 
plane angle, x = 12° 20', and what velocity has it at 38 feet from the starting 
point. 

~ \/ 3M6xSl 2P W ’ 

F = g Thin.x = 32T6X3‘33Xsin , 12 o 20 / = 22-8 feet per second. 

Power Concentrated in Moving Bodies* 

It is highly important to distinguish betweeu power simply, and power when 
concentrated in a moving body. The former is the force multiplied by its velo¬ 
city, —but the power concentrated in a moving body is equal to the weight of the 
body multiplied by the square of its velocity, and the product divided by the 
accelleratrix g, —or the power concentrated in a moving body is equal to the 
power expended in giving it the motion. 

Example . A sledge weighing 20 pounds, strikes a nail with a velocity of 12 
feet per second. With what effect did it strike? 

P - - 89-55 effects. 


L 



















Force op Gravity. 


185 




























































186 


Centrifugai Force, 


CENTRIFUGAL FORCE. 


Central Forces are of two kinds, centrifugal and centripetal . 

Centrifugal Force is the tendency which a revolving body has to 
depart from its centre of motion. 

Centripetal Force is that by which a revolving body is attracted or at¬ 
tached to its centre of motion. 

The Centrifugal and Centripetal forces are opposites to each other, and when 
equal the body revolves in a circle; but when they differ the body will revolve 
in other curved lines, as the Ellipse, the Parabola, &c., according to the nature 
of the difference in the forces. If the centrifugal force is o while the other is 
acting, the body will move straight to the centre of motion ; and if the centripe¬ 
tal force is o while the other is acting, the body will depart from the circle in a 
straight line, tangent to the circle in the point where the centripetal force ceased 
to act. The central forces are distinct from the force that has set the body in 
motion. 

If the centrif ugal force be made use of to produce an effect, such effect will be 
at the expense of the one producing the rotary motion. 

Letters denote. 

F = Centrifugal force in pounds. 

M = the Mass or weight of the revolving body in pounds, 
v = Velocity of the revolving body, in feet per second. 

R = Radii of the circle in which the body revolves, in feet. 
rt = number of revolutions per minute. 

Example 1. Required the centrifugal force of a body weighing 63 pounds, and 
making 163 revolutions per minute, in a circle of 4 feet, 4 inches radius? 


M R 
2933 


63X4-33X1639 

2933 


= 2475 pounds. 


Example 2. 
115 feet radii. 


A Railroad train runs 43 miles per hour on a curved track of 
What should be the obliquity of the track ? 


tan.# = ■ 


Miles* 43* 

6 2iR' ~ 69X115 — °’ 233 * 
or x = 13° 10', the obliquity of the track. 

Example 3. A governor having its arms Z = 1 foot, 6 inches, how many revol¬ 
utions must it make per minute to form an angle x — 30° ? 

54-16 

n = -—— = 47*5 revolutions per minute. 

>/l-5Xcos.30 o j 



1 , 

2, 

3 , 

4 , 

5 , 

































Centrifugal Force Governors. 


187 



228. 


Centrifugal force of a ring. 
M n* V Ra+'^a 


F = 


4150 



229. 

Centrifugal force of a grinding stone, 
circle-plane, cylinder , rotating round 
this centre. 




MRn * 
4150 ‘ 



230. 

Centrifugal force of a cylinder rotating 
round the diameter of its base. 


p - Mn 3 V~4~F+Sr 

~ 10260 



231. 


Centrifugal force of a ball, 
(centre of gyration included.) 


F=* 


M m \T R 2 +|r* 
"2933 » 



232. 


6o n 
n “2 h\Jl 


Governor. 

9 54-16 _ 54-16 

Vh V l cos x ’ 


h = 


2933 


7 = 


2933 


n cos.# cos.# 


2933 h 


bos.*-^£-V, r=VF^ 

if I l 





























Pendulum. 


188 ' 


PENDULUM. 


Simple Pendulum is a material point under the action of gravitation, 
and suspended at a tixed point by a line of no weight. 

Compound Pendulum is a suspended rod and body of sensible mag¬ 
nitude, tixed as the simple pendulum. 

Centre of Oscillation is a point in which if all the matter in the com¬ 
pound pendulum were there collected, it would make a simple pendulum oscil¬ 
late at the same times. 

Angle of Oscillation is the space a pendulum describes when in mo¬ 
tion. 

The velocity of an oscillating body through the vertical position, is equal to 
the velocity a body would obtain by falling vertically the distance versed sine of 
half the angle of oscillation. 

Letters denote. 

I = length of the simple pendulum, or the distance between the centre of sus¬ 
pension, and centre of oscillation in inches. 

t — time in seconds for n oscillations. 

n = number of single oscillations in the time t. 

Example 1. Required the length of a pendulum that will vibrate seconds? 
here n = 1, and t = 1". 

39*109 inehes, the length of a pendulum for seconds. 

Require the length of a pendulum that will make 180 vibrations 
here t = 60" and n — 180. 


I = 39T09 — 
n a 

Example 2. 
per minute? 


I = 


39T09? 3 39-109X 60 3 


- 4*346 inches. 


Example 3. 
in 8 seconds ? 


n 3 iso 3 

How many vibrations will a pendulum of 25 inches length make 

6-254? 


n = 


yt 


6*254X3 

+25“ 


10 vibrations. 


Example 4. A pendulum is 137*67 inches long and makes 8 vibrations in 15 
seconds. Required the unit or accelleratrix g — ? 


9 


0-8225? ?i3 0-8225Xl37-67XS ! » 


?a 


153 


= 32-209. 


Example 5. A compound pendulum of two iron balls P and Q, having the 
centre of suspension between themselves: see Fig. 238. P = 38 pounds, Q = 12 
pounds, a = 25 inches, and b = 18 inches. How long is the simple pendulum, 
and how many vibrations will the pendulum make in 10 seconds? 


I = 


a P — b Q 25X38 — 18X12 
JP+Q 

Ct3 P+62 Q 


38+12 

253X38+183X12 


14*68 inches. 


x(P+Q) 14*68.(38+12) 

the length of the single pendulum. 

6-254? 6-254X10 


= 37*68 inches, 


yt 


+37*68 


— 10-193 vibrations in 10 seconds. 


If a compound pendulum is hung up at its centre of oscillation, the former 
centre of suspension will be the centre of oscillation, and the pendulum will 
oscillate the same time. 


I 





















Pendulum and Centre op Oscillation. 


189 


/ 

1 

1 

« 

1 

\ 

*-•$> 

.9 

u -7 1 

A A 

z 

*4 

A 

-y 

B 

__V ' 


£ 

V- 



233. 

Simple ’Pendulum. 
I = I2<rp = 39-1/ 9 

Tt 1 n* if 

n\f l 
6.25’ 

6 254 1 

n= f/T’ 



34. 

A = centre of grav¬ 
ity. 

B = centre of gyra¬ 
tion. 

C = centre (f oscil¬ 
lation. 

a : 5 = b ; /, 

6= v/77=l-1432a, 
/ = Ua. 


235. 

Compound Pendu¬ 
lum. 

r = radius of cylin¬ 
der. 


16a 2 -)- 3r 9 

12a 


4a r 9 

'" t + k - 


a/ 





236. 

/?T 2 

* = 12F ’ 

0-8225/n 9 

O- _- 

b t 9 ’ 

o = centre of suspen¬ 
sion. 

i 2r 

/ = a 4- r —. 

5 a 


237. 


7 _ -P+^ 9 Q 

~ «P+5 Q ' 

P and Q expressed 
in pounds, or cubic 
contents. 


238. 


x = 


aP — bQ 
P+Q ’ 


, _ a 2 P+/> 9 Q 
~ lc{PfQY' 


Length of a Pendulum vibrating seconds at the level of the sea, in various places. 

At the Equator, lat. 0° O' 0" 

“ Washington, lat. 38° 53' 23" 

“ New York, lat. 40° 42' 40" 

“ London, lat. 51° 31' - - - 

“ lat. 45° - 

« Stockholm, lat. 59° 21' 30" - 

l = 39-127 — 0-09982 cos.2 lat. for seconds. 


39-0152 inches. 
39-0958 
39-1017 
39-1393 
39-1270 
39-1845 
























































190 


Centre of GyAation. 


CENTRE OF GYRATION. 

Centre of Gyration is a point in revolving bodies in which, if all the re¬ 
volving matters were there contained, it would obtain equal angular velocity 
from, and sustain equal resistance to, the force that gives it a rotary motion. 

The centre of gyration in different bodies will be found by the accompanying 
formulas, in which x = distance from the centre of motion to the centre of gyra¬ 
tion. 

Example 1. Fig. 239. Find the centre of gyration in a bar, rotating round one 
of its ends; its length is 7 feet, 3 inches? 

x = 0*5775X7*25 — 4T3 feet, from the centre of motion. 

Example 2. Fig. 245. Find the centre of gyration of a cone, rotating round its 
vertex, its height being h = 3*3 feet, and R = 8 inches = 0*666 feet. 




/ 12h*+3R* f 

V 20 v 


12X3*3^+3X0*666^ = 9/fifiQ w 
20 



from the centre of motion. 

Example 3. Fig. 249. A ring or fly wheel having its outer radius R = 6 feet 
4 inches, the inner radius r — 5 feet 8 inches. Required its centre of gyration 
x — ? from the centre of motion. 


= 6 feet 


CONCLUSIONS. 

The object of finding the centre of gyration of revolving bodies is to ascertain 
what effect is necessary to give a mass a certain angular velocity; or how much 
effect is concentrated in a body having a certain angular velocity. 

Angular velocity is the number of revolutions a body makes in a unit of time, 
it is herein denoted by the letter n. 

Letter js denote. 

P = power in effects. 

H = horse-power. 

F = the Force which is applied to rotate a body, in pounds. 

s = the radius on which the force acts, in feet. 

M = Mass of the revolving body, in pounds. 

x = the distance from the centre of motion to centre of gyration, in feet. 

T = time the force F is applied in seconds. 

jV = number of revolutions in the time T. 

n ,= angular velocity or number of revolutions per minute, at the end of the 
time T. 

g = 32*166 accelleratrix of the force of gravity. 

G = accelleratrix of the force F, then, 


G : g — F S* : Mx*, or G — 


g FS 9 
M x a ’ 


Example 4. Fig. 249. In connection with the preceding example (3) the fly-wheel 
weighs 7400 pounds. "W hat force F must be applied at the radius r = 2 feet, to 
give the fly wheel an angular velocity of n = 128 revolutions per minute, at the 
end of the time T = 40 seconds ? 


Formula 6. 


n Mx* 128X7400X'6 a 


2773 pounds. 


F — 153*5pr TS — 15+5X40X2 
How many revolutions did the wheel make in the 40 seconds ? 
2*56 To. FS 


Formula 9. 


N - 


2*56x40 2 X2773X2 or , 

- shwv. ' ^ -- = 85*27 revolutions. 

7400XS+ 
























Centre of Gyration. 191 






















































192 Centre of Gyration. 











































Centre of Gyration. 


193 


T = 


Formulas for Force and Power of Acceleration* 

2-56 T'F s 


sP 


4/r 5 N 


G ’ 




47t N M 


g F s 


T = 


2 Tt s n 


T = 


60G ’ 

4rc n M x* 


60^ F s ’ 


T1 /V M x* 

F = 


2*56 T 2 5 ’ 


P- 


n M x* 


s = 


153*5 T s' 

N M x 2 


5 = 


2-56 T 2 P’ 

n M x* 


153*5 TP’ 


1, 

2 , 

3 , 

4 , 

5 , 

6 , 

7 , 

8 , 


iV = 


n = 


153-5T P 5 


P = 


fi 2 iVf # 2 




244 T ’ 
244 TP 


p M X % 7? 


244P ' 


n 


v 


244 TP 


iVf x 2 


H = 

M = 


ri 1 M x 2 


134100T’ 
134100T H 


- 9, 

* 10, 

- 11 , 
- 12 , 

- 13, 

- 14, 

- 15, 
16, 


Fly-Wheels. "Weight of. 


The weight of a fly-wheel will be determined by the formula 16 in which the 
time T = 130 seconds, the time in which the fly-wheel would concentrate the 
same power as the steam-engine. When the works or resistance is very irre¬ 
gular it will be better to take the time T = 170. The centre of gyration (in¬ 
cluding ring and arms,) can in practice be assumed at x — r the inner radius of 
the ring. 


Example 5. Required the weight of a fly-wheel for ordinary work, the steam 
engine being 56 horse power, making 42 revolutions per minute, and the inner 
radius r = 10 feet ? 


„ 134100TB’ 134100X130X56 

-- 42 , x i 0 g — “ 6535 P° unds * 


n 4 


17 






























194 


Centre of Gravity. 


CENTRE OE PERCUSSION. 

Centre of Percussion is a point in which the momentums of a moving 
body are concentrated. Centre of Percussion is the same as centre of oscillation, 
and to be calculated by the same formulas. 

Take an iron bar in one hand, and strike heavily over a sharp edge, if the 
centre of percussion of the bar strikes over the edge, the whole momentum will 
there be discharged, but if it strikes at a distance from the centre of percussion a 
part of the momentum will be discharged in the hand, and a shock felt. 

It is sometimes of great importance to properly place the centre of percussion 
Tf it is dislocated, the moving body not only fails to properly transmit its effect, 
but the lost momentum acts to wear out the machinery. 


♦ 4 


CENTRE OF GRAVITY. 

Centre of Gravity is a point around which the momentums of all matters 
funder tire action of the force of gravity) in a body, or system of bodies, are 

A body or system of bodies suspended at its centre of 
gravity, will be in equilibrium in all positions, 

A body or system of bodies, suspended in a point out of 
its centre of gravity, will hang with its centre of gravity ver¬ 
tical under the point of suspension. 

A body or system of bodies suspended in a point out of 
its centre of gravity, and having two different positions, 
the two vertical lines through the point of suspension 
will meet in the centre of gravity ; thus if a plane be hung 
up in two different positions, the vertical lines a, b, and 
c, d, will meet in the centre of gravity o. 

z = distance to the centre of gravity as noted in the 
figures. 

Example, 1. The radius of a circle being 3 feet, how far is 
its centre of gravity from the centre of the half circle ? 

^ = 0-6367 X 3 = 1*91 feet. 

Example 2. How far from the bottom of a cylindric shell, 
open at one end, is its centre of gravity ? The cylinder is 
4 feet long, radius r = 0-8 feet. 




h __ 4 

r+2ft 0-8+2X4 


= 0-625 feet. 


Example 3. Fig. 264. An irregular figure weighing P = 138 pounds, is sus¬ 
pended between a fulcrum and a weight, l = 5"6 feet, IF— 57 pounds. Re¬ 
quired the distance to the centre of gravity z — ? 















Centre of Gravity. 


Quadrangle .—a and b 'parallel, 
h h /b — au 
Z ~ 2 6 ' 6+a) '* * 


255. 


Circle sector. 
2c r 
Z= "'3 b m 


Half a circle plane or Elliptic plane, 
z = 0424r. 


Circle Segment, a — area. 
c 3 

* ~ 12a* 

x = h\z — r. 


25/. Parabola. 

2 h 


For half a Parabola x = g b. 


256. 

























196 


Centre op Gravity. 



258. 

Half Sphere. 

Convex surface . . . z = £r. 

Solid. 2 — §r. 

259. 

Spherical Sector. 

Solid, 





260. Spherical Segment. 
Convex surface z = 

A 


Solid 


h p2rM-A 9 "| 


^ -*, 


261. 


Cone. 



Convex surface 2 = —, 


Solid 


Z ~ 4" 



Come Fustrum. 

A ftr#-r 


262. 

A ft r i? — r 1 
Con. sur. * - g- - g[ mr -J 


Solid 


z _h l&+t(2R-rZr) 


R* i r(R+r) 


263. 



1 yr ami die Fustrum. 


A and a — area of the two bases. 

h rA < 3a+2\/ A a 1 


Solid z = 


ifa+\/ A a 























Cents,*, of Guvmr. 


197 



































198 
I- 


Specific Gravity. 


SPECIFIC GRAVITY. 

Specific Gravity is the comparative density of substances. The unit for 
measuring the specific gravity is assumed to be the density of rain water, or 
distilled water. 

One cubic foot of distilled water weighs 1000 ounces, or 62'5 pounds avoir¬ 
dupois. 

To Find the Weight of a Body, 

RULE 1. Multiply the contents of the body in cubic feet by 62 - 5, and the 
product by its specific gravity, will be the weight of the body in pounds 
avoirdupois. 

RULE 2. Multiply the contents of the body in cubic inches by 0-03616, 
and the product by its specific gravity, will be the weight of the body in 
pounds avoirdupois. 

RULE 3. Divide the specific gravity by 0-016 and the quotient is the weight 
of a cubic foot. 

Example 1. A bottle full of mercury is 3 inches, inside diameter, and 6 inches 
high. How much mercury is there in the bottle in pounds? 

One cubic inch of mercury weighs 0-491 pounds, and by the formula for 
Fig. 119 we have the 

weight = 0-49lX0'785X3 2 X6 = 20-85 pounds. 

Example 2. Required the weight of a cone of cast iron, diameter at the 
base d = 1-33 feet, height h = 4 feet? One cubic foot of cast iron weighs 
450-5 pounds, and by formula for Fig. 117 we have the 

weight = 45Q-5X0'2616Xl'33 2 X4 = 834 pounds. 

Example 3. The section area of the lower hole in a steam boat is 245 square 
feet; how much space must be taken in the length of the hole for 131 tons 
of anthracite coal? 

Anthracite coal are 42-3 cubic feet per ton. 

42-3 vl 31 

length = ——— = 22"6 feet, the space required. 


Weight and Bulk of Substances* 




Cubic 

fed 

Cubic 

foot 

Names of Substances. 

Sand, ... 

Granite, - 
Earth, loose, - 
Water, salt, (sea) - 
“ fresh - 

Ice, - 

Gold, ... 

Silver, ... 

Cubic 

feet 

Cubic 

foot 

XI \JJ AJCt/L Ol'H/H’OOt 

Cast iron, 

Wrought iron, 

Steel, ... 

Copper, - 

Lead, - 

Bi-ass, - 

Tin, - - - 

Pine, white 

in 

pounds. 

450-5 

486-6 

489-8 

555* 

707-7 

537-7 

456 

2956 

per 

ton. 

4-97 

4-60 

4-57 

4-03 

3- 16 

4- 16 
4-91 
75-6 

in 

pounds. 

94-5 

139 

78-6 

64-3 

62-5 

58-08 

1013 

551 

per 

ton. 

23-7 

16-1 

28-5 

34- 8 

35- 9 
38-56 
2-21 
4-07 

“ yellow, - 


33-81 

66-2 

Coal, Anthracite 

53 

42'3 

Mahogany, 


66-4 

33-8 

“ Bituminous - 

50 

44-8 

Marble, common, 
Mill-stone, 


141-0 

15-.9 

“ Cumberland - 

53 

42-3 


130 

17-2 

“ Charcoal 

18-2 

123 

Oak, live - 
“ white, 

Clay, 


70 

32-0 

Coke, Midlothian - 

32-70 

6S-5 


45.2 

101-3 

49-5 

22-1 

“ Cumberland - 
“ Natural Virginia 

31-57 

46-64 

70-9 

48-3 

Cotton Bales, - 
Brick, 

Plaster Paris, - 


100 

105 

22-4 

21-3 

Conventional rate of 
Stone coal, 28 bushels 
(5 pecks) = 1 ton. - 


43-56 | 



















Specific Gravity. 


199 


To Find tlio Specific Gravity* 

W — weight of a body in the air. 

w = weight of the body (heavier than water) immersed in water. 
S — specific gravity of the body. Then, 

W — w : W= 1: & S = ■—, .... 

W — U)’ 


1, 


Example. 4. Required the specific gravity of a piece of iron-ore weighing 
6-346 pounds in the air, and 4-935 pounds in water, S = l 

S 6-345 6 —4-935 = 4 ‘ 5 the s P ecific gravity. 

When the body is lighter than water, annex to it a heavier body that is able 
to sink the lighter one. 

S == specific gravity of the heavier annexed body, 
s = specific gravity of the lighter body. 

W = weight of the two bodies in air. 
w = weight of the two bodies in water. 

V = weight of the heavier body in air. 
v == weight of the lighter body in air. 


V 3 

W — w - 

S 


2 , 


Example 5. To a piece of wood, which weighs t> = 14 pounds in the air, is 
annexed a piece of cast-iron V == 28 pounds; the two bodies together weigh 
w = 11-7 pounds in water. Required the specific gravity of the wood? 

W— V-T-v = 28+14 = 42 pounds. 

S = 7'2 specific gravity of cast-iron. 


Formula 2. 


*S = 


14 


42 — H-7— £■ 
7-2 


753 - = 0-529, the specific 
Zo 


gravity of the wood, (Poplar White Spanish.) 

A simple way to obtain the specific gravity of woods, is to form it to a parallel 
rod, and place it vertically in water, then when in equilibrium, the immersed 
end is to the whole rod as the specific gravity is to 1 *. 

Example 6 . A cylinder of wood is 6 feet, 3 inches long, when Immersed verti¬ 
cally in water it will sink 3 feet, 9 inches by its own weight. Required its spe¬ 
cific gravity. 

3-75 : 6-25 = #:1, S= 0-600. 

6’25 

To discover the Adulteration in Metals, or to find the proportions of two Ingredients 

in a Compound. 

W — s( W — w) 

K= ■ > “ * * - * • o, 

1— ~S 

Example 7. A metal compounded of silver and gold weighs W— 6 pounds 
in the air, and in water w — 6-636 pounds. Require the proportions of silver 
and gold ? 

S = 19-36 specific gravity of gold. 
s — 10-51 specific gravity of silver. 

weight V == — 4 0.51(6 - 0.6 36 ) _ pounds of gold. 

10-51 


1 — 


19-36 


and 1-245 pounds of silver. 









200 


Specific Gravity. 










'-^- 




Weigh 




} 

Specific 

gravity. 

Weight 

Names of Substances. 

Specific 

gravity. 

per 

cubic 

Names of Substances. 

per 

cubic 



inch. 





inch. 

Metals. 









Platinum, rolled - 

- 

22-669 

•798 

Alabaster, white 

- 

- 

2-730 

•0987 

“ ' wire, ‘ - 


21-042 

•761 

“ yellow 

- 

- 

2-699 

•0974 

“ hammered. 

20-337 

•7S6 

Coral, red - - - 

- 

- 

2-700 

•0974 

“ purified, 


19-50 

•706 

Granite, Susquehanna 

2-704 

•0976 

“ crude, grains 

15-602 

*665 

. “« Quincy 

- 


2-652 

*0958 

Gold, hammered - 

- 

19-3G1 

•700 

“ Patapseo 

- 


2-640 

•0954 

“ pure cast - - 

- 

19-25S 

•697 

“ Scotch - 

- 


2-625 

•0948 

“ 22 carats fine 

- 

17-486 

•733 

Marble, white Italian 

2-708 

•0978 

“ 20 “ “ 

- 

15-702 

•568 

“ common 

- 


2-686 

•096S 

Mercury, solid at — 

40° 

15-632 

•566 

Tale, black - - 

- 


2-900 

•0105 

“ at 4-32° Fahr. 

13-619 

•493 

Quartz, - - - - 



2-660 

•0962 

“ « so° 

U 

13-580 

•491 

Slate, - - - - 

- 


2-672 

•0965 

ee (c 212° 

a 

13*375 

•484 

Pearl, oriental - 

- 


2-650 

•0957 

Lead, pure ... 

- 

11*330 

•410 

Shale, - - - - 



2-600 

•0940 

“ hammered - 

- 

11*388 

•412 

Flint, white - - 

- 


2-594 

•0936 

Silver, hammered - 

. 

10-511 

•381 

“ black - - 

- 


2-582 

•0933 

“ pure - - - 

. 

10*474 

•379 

..'tone, common - 

- 


2-520 

•0910 

Bismuth, - - - - 
Red Lead, - » - 

- 

9-823 

8*940 

•355 

“ Bristol - 
“ MHI - - 

- 


2-510 

2-484 

•0906 

•0897 


*o24 



Cinober, - - - - 

- 

8-098 

•293 

“ Paving - 

- 


2-416 

•0873 

Manganese, - - - 

- 

8-030 

•290 

Gypsum, opaque 

- 


2-168 

•0783 

Copper, wire and rolled 

8-878 

•321 

Grindstone, - - 

- 


2-143 

•0775 

“ pure - - - 

- 

8-788 

•318 

Salt, common - 

- 


2-130 

•0770 

Bronze, gun metal 

- 

8-700 

•315 

Saltpetre, - - - 



2-090 

•0755 

Brass, common - - 

- 

7-820 

•282 

Sulphur, native 

- 


2-033 

•0735 

Steel, cast steel - - 

- 

7-919 

"286 

Common soil, - 

- 


1-984 

•0717 

“ common soft 

- 

7-833 

•2S3 

Rotten stone, 

- 


1-981 

0416 

“ hardened & temp. 

7 -sis 

•283 

Clay, - - - - 



1-930 

•0698 

Iron, pure - - - 

- 

7-768 

•281 

Brick, - - - - 



1-900 

•0686 

“ wrought and rolled 

7-780 

•282 

Nitre, - - - - 

- 


1-900 

•0636 

“ hammered - 
“ cast-iron - - 

: 

7-789 

7-207 

•282 

•261 

Plaster Paris, - 

- 

{ 

1- 872 

2- 473 

•0677 

•0894 

Tin, from Bohmen 


7-312 

•265 

Ivory, - - - - 



1-822 

•0659 

“ English - - - 

- 

7-291 

•264 

Sand, - - - - 



1-800 

•0G51 

Zinc, ro’led - - - 

- 

7-191 

•260 

Thosphorus, - - 

- 

- 

1-770 

•0640 

“ cast --- - 


6-861 

•248 

Borax, - - - - 



1-714 

•0620 

Antimony, - - - 
Chrom, ----- 

• 

6-712 

5-900 

•244 

•213 

Coal, Anthracite 

- 

/ 

i 

1-640 

1-436 

*0593 

•0592 

Arsenic, - - - - 


5-763 

•20S 

“ Maryland - 

- 


1-355 

•0490 ‘ 

Stones and Earths. 



“ Scotch - - 

- 

- 

1-300 

•0470 

Topaz, oriental 




f* New Castle 

. 

. 

1-270 

•0460 

- 

4-011 

•145 

“ Bituminous 

- 

. 

1-270 

•0460 

Emery, ----- 
Diamond, - - - - 


4-000 

3-521 

•144 

•127 

Charcoal, triturated 
Earth, loose - - - 

- 

1-380 

1-500 

*0500 

*0542 

Limestone, green - 

- 

3-180 

•115 

Amber, - - - - 



1-078 

‘0387 

white - 

Asbestos, starry 
Glass, flint - - - 

- 

3-156 

3-073 

2-933 

•114 

•111 

•106 

Pimstone, - - 
Lime, quick - - 
Charcoal, - - - 

“ 

* 

1-647 

0-804 

0-441 

•0596 

•0291 

•0160 

“ white - - - 

. 

2-892 

•104 




“ bottle - - - 

. 

2-732 

•09S7 

Woods (Dry.) 



“ green - - - 

- 

2-642 

•0954 

Alder, - - - - 



•800 

*0289 

Marble, Parian - - 

- 

2*838 

•103 

Apple-tree, - - 


- 

•793 

•0287 

“ African 

- 

2-708 

•0978 

Ash, the trunk - 


- 

•845 

•0306 

“ Egyptian - 

- 

2-668 

•0964 

Bay-tree, - - - 


- 

•822 

•0297 

Mica, ----- 


2-S00 

•1000 

Beech, - - - - 



•852 

•0308 

Hone, white razor 

. 

2-838 

-104 

Box, French - - 


. 

•912 

•0330 

Chalk,. 

- 

2-784 

•100 

“ Dutch - - 



1-328 

•0480 

Porphyry, - - - - 


2-765 

•0999 

“ Brazilian red 


. 

1-031 

•0373 

Spar, gr^en - - - 

- 

2-704 

•0976 

Cedar, wild - - 


- 

•596 

•0219 

•‘ blue - - - 

- i 

2-693 

■0971 

“ Palestine 


- 

•613 

•0222 



n._t 





































Specific Gravity. 


201 


Names of Substances . 

Cedar, Indian - 
“ American 

Citron, - 
Cocoa-wood, 

Cherry-tree, 

Cork, - 

Cypress, Spanish - 
Ebony, American - 
“ Indian 

Elder-tree, 

Elm, trunk of - 
Filbert-tree, 

Fir, male - 
“ female 

Hazel, - 
Jasmine, Spanish - 
Juniper-tree, - 

Specific 

gravity. 

1-315 

•561 

•726 

1-040 

•715 

•240 

•644 

1-331 

1-209 

•695 

•671 

•600 

•550 

•498 

•600 

•770 

•556 

Weight 

per 

cubic 

inch. 

•0476 

•0203 

■0263 

•0376 

•0259 

•0087 

•0233 

•0481 

•0437 

•0252 

•0243 

•0217 

•0199 

•0180 

0217 

•0279 

•0201 

Lemon-tree, 

•703 

•0254 

Lignum-vitae, - 

1-333 

•0482 

Linden-tree, - 

•604 

•0219 

Log-wood, 

•913 

•0331 

Mastic-tree 

•849 

•0307 

Mahogany, 

1-063 

•0385 

Maple, - 

•750 

•0271 

Medlar, - 

•944 

•0342 

Mulberry 

•897 

•0324 

Oak, heart of, 60 old 

1-170 

•0423 

Orange-tree, - 

•705 

.0255 

Pear-tree, 

•661 

•0239 

Pomegranate-tree, - 

1-354 

•0490 

Poplar, - 

•383 

•0138 

“ white Spanish 

•529 

•0191 

Plum-tree, 

•785 

•0284 

Quince-tree, 

•705 

•0255 

Sassafras, 

*482 

•0174 

Spru e, - 

*500 

•0181 

" old 

•460 

•0166 

Pius, yellow - 

•660 

•0239 

“ white 

•554 

•0200 

Vine, - 

1-327 

•O4S0 

Walnut, - 

•671 

•0243 

Yew, Dutch 

•788 

•0285 

“ Spanish - 

•807 

•0292 

Liquids* 



Acid, Acetic - 

1-062 

•0384 

“ Nitric 

1*217 

•0440 

t( Sulphuric 

1-841 

•0666 

“ Muriatic 

1-200 

•0434 

M Fluoric - 

1-500 

•0542 

il Phosphoric - 

1-558 

•0563 

Alehohol, commercial 

•833 

•0301 

“ pure 

•792 

•0287 

Ammoniac, liquid - 

•897 

•0324 

Beer, lager 

1-034 

•0374 

Champagne, - 

9-97 

•0360 

Cider, - 

1-018 

•0361 

Ether, sulphuric 

•739 

•0267 

Egg, - - - - 

1-090 

•0394 

Honey, - 

. 1-450 

•0524 

Human blood, 

1-054 

•0381 

Milk, 

1-032 

•0373 




Weight 

Names of Substances. 

Specific 

gravity. 

per 

cubic 



inch. 

Oil, Linseed - 

•940 

•0340 

“ Olive - 

•915 

•0331 

“ Turpentine 

•870 

•0314 

“ Whale 

•932 

•0337 

Proof Spirit, 

•925 

•0334 

Vinegar, - 

1-080 

•0390 

Water, distilled 

1-000 

•0301 

Sea 

1-026 

•0371 

“ Dead sea 

1-240 

•0448 

Wine, ... 

•992 

•0359 

“ Port 

•997 

•0361 

Miscellaneous. 



Aspbaltum, - j 

•905 

1-650 

•0327 

0597 

Beeswax, - 

•965 

'0349 


•942 

'0341 

Camphor, 

•988 

*0357 

India rubber, - 

•933 

•0338 

Fat of Beef, 

•923 

’0334 

“ Hogs, - 

•936 

•0338 

“ Mutton, 

•923 

•0334 

Gamboge, 

1-222 

•0442 

Gunpowder, loose - 

•900 

•0325 

“ shaken 

1-000 

•0361 

“ solid - | 

1-550 

•0561 

1-800 

•0650 

Gum Arabic, - 

1-452 

•0525 

Indigo, - 

1-009 

•0365 

Lard, - 

•947 

•0343 

Mastic, - 

1-074 

•0388 

Spermaceti, 

•943 

•0341 

Sugar, - 

1-005 

- 05S0 

Tallow, sheep - 

•924 

•0334 

“ calf 

•934 

•08i>8 

“ ox, ‘ - 

•923 

•0334 

Atmospheric air, 

•0012 

.43 


Weight 

Gases* Vapours* 


cub. ft. 
grains. 

Atmospheric air, 

1-000 

527-0 

Ammoniacal gas, - 

•500 

262-7 

Carbonic acid, - 

1-527 

805-3 

Carbonic oxid, 

•972 

512-7 

Carburetted hydrogen, 

•972 

512-7 

Chlorine, - 

2-500 

1310 

Chloroearbonous acid, 

3 “47 2 

182S 

Chloroprussic acid, 

2-152 

1134 

Flouborie acid, 

2-371 

1250 

Hydriodic acid, 

4-346 

2290 

Hydrogen, 

•069 

36*33 

Oxygen, - 

1104 

581-8 

Sulphuretted hydrogen, 

1-777 

9370 

Nitrogen, 

Vapour of Alehohol, 

•972 

512-0 

1-013 

851-0 

“ turpen’e spir., 

5-013 

2042 

“ water, 

•023 

328-0 

Smoke of bitumin. coal, 

•102 

53-S0 

“ wood, 

•90 

474-0 

Steam at 212° - 

•4S8 

257 3 


































202 


Hydrometer. 


HYDROMETER. 

A eody wholly immersed in a liquid will lose as much of its weight, as the 
weight of the liquid it displaces. 

A floating body will displace its own weight of the 
liquid in which it floats. 

A cylindrical rod of wood or some light materials, 
being set down in two liquids, A and B, of different 
specific gravities, when in equilibrium it will sink to 
the mark a in the liquid A, and to b in the liquid B; 
then the specific gravity of A : B — b, c : a, c, or in¬ 
verse as the immersed part of the rod. This is the 
principle upon which a hydrometer is constructed. 


Table showing the comparative Scales of Guy Lussac and Baume, with the Specif <: 
Gravity and Proof at the temperature of 60° Fahr. 



270. n 

ft 


An 


VAC 


-a 


b 




HYDROST ATICS. 

Letters denote. 

A and a = areas of the pressed surfaces in square feet. 

P and p = hydrostatic pressure in pounds. 

d — depth of the centre of gravity of A or a under the surface of the liquids 
In feet. 

S = specific gravity of the liquid. 

Example 1. Fig. 272, The plane A = 3-3 square feet, at a depth of d — 6 feet 
under the surface of fresh water. Required the pressure P = ? Specific gravity 
of fresh water S = 1. 

P = 62-5A d = 62*5X3-3X6 = 1237-5 pounds. 

Example 2. Fig. 275. The area of the pistons A = 8-5 square feet, a = 0 - 02 
square feet, l = 4 feet, e = 9 inches, and F = 18 pounds. Required the pres¬ 
sure P — ? , 


P = 


FI A 

-e a 


18X4X3-5 

0-75X0*02 


= 40800 pounds. 


It must he distinguished that the centre of pressure and centre of gravity of 
the planes, are two different points; the centre of pressure is below the centre 
of gravity, when the plane is inclined or vertical. 


.j 




























































IItdjiostatics. 


20" 


272. 



P - G2-5 &A r d, 

p _ 
62 : 5*S’ d, 

P 


4 = 


d - 


62-5 S -A.* 



273. 


TAe Hydrostatic paradox. 


The pressure P is independent of the 
width of column C. 

P = 62*5 S A h. (same as above.) 



P - .4(62-55; A + £) 
p = a (£-62-5SA), 


A - 


P a - p A 
( 2*5 <SA a 






275. Bra nali s Hydraulic Press. 


P ~ FlA -, 

e a 


t\ Pea 

F = —, 


. Pea 


A 


a ----- 


P AJ 

~P e 


276. Centre of Pressure of a rectangle , 
the upper edge at the surface 

of the liquid d = § A. 

277. Centre of Pressure of a triangle, 
the base being at the surface of 

the liquid , d *=\h. 



278. Centre of Pressure of a 
triangle, the vertex being at the surface 

of the liquid, d •= * h. 

279. 


3 + V r 4( A-// f 


A’. 
















































204 


Stabit itt or Floating Bodies. 


STABILITY OF FLOATING BODIES. 

We have before said that a floating body displaces its own weight of the liquid 
in which it floats. 

A floating body is in equilibrium when its centre of gravity is in the same 
vertical line, as the centre of gravity of its displacement. 

The momentum of stability of a floating body, is equal to the weight of the dis¬ 
placement, multiplied by the horizontal distance between the two centres of 
gravity of the body and the displacement. 

Metacentrum is the point where the vertical line drawn through the centre of 
gravity of the body when in equilibrium, crosses the vertical line drawn through 
the centre of gravity of the displacement when the body is out of equilibrium. 

When the centre of gravity is below metacentrum, the momentum of stability 
is positive, but negative when above. 

A boat will capsize when its centre of gravity passes metacentrum. 

Letters denote. 

L, B and d = Length, Breadth, and depth in feet, of the displacement, when 
the floating body is a parallolipipedon. 

Q = weight of the displacement of a vessel, in pounds. 

P — weight of a cubic foot of the liquid in which the vessel floats, in pounds. 

a = distance between the two centres of gravity of the vessel and the dis¬ 
placement when in equilibrium, in feet. 

b = the horizontal distance between the same centres of gravity, in feet. 

F= the force which acts to careen the vessel out of equilibrium, in pounds. 

I = the lever on which the force F acts, in feet, measured from the centre 
of gravity of the displacement at right-angles to the direction of the force F. 

FI = momcn um of stability. 

c = centre of gravity of the vessel. 
m — metacentrum. 

= the greatest immersed section area of the vessel, in square feet. 

Example 1. Fig. 281. A vessel being of the dimensions L = 20 feet, B = 7 feet 
3 inches, d — 3 feet. Distance between the centres of gravity — a = 8 inches, 
h — 4 feet. Required the momentum of stability in fresh water, FI = ? 
P = 62-5. 

FI = PL d = 62-5X20X3X2X4^||^ - 0*666^ 

= 23820 momentums of stability. Let the force be applied on a lever l — 18 
feet. Required the force F = l 

F = ~= 1326'6 pounds. 

Example 2 Fig. 283. What force (P = ?) is required to careen a vessel of 800 
tons? the greatest immersed section J$ = 314 square feet. B = 28 feet beam, 
and the distance between the centres of gravity-fa — 1*4 feet. To be careened 
to an angle v = 30°. 

FI = Q sin.r(^-fa) = 800X2240X^.30^^+1*4^ 

= 5710480 momentums of stability. 

The lever on which the force F acts l = 25 feet. 

Then, P= = 228659*2 pounds,—102 tons. 

In the same manner the careen can be calculated by the force of wind on the 
sails. 






Momentum of Stability. 20b 
























































































206 


Hydraulics. 


HYDRAULICS. 

Let the vessel A, Fig. 284, be kept constantly full of water up to the water 
line tv. In two horizontal faces lower than the water line tv, are made orifices 
a and a', through which the water will pass up vertical nearly to the water 
line w. Omitting the resistance of air, &c., the jet should theoretically reach 
the water line w; practically it reaches 0'967 h. 

It is evident that the velocity of the jet through the orifices, must he the ve¬ 
locity due to a body falling the height h, according to the law of force of 
gravity. 

Letters denote. 

Q = actual quantity of water discharged per second or in the time t, in cubic 
feet. 

h — head, or height of water over the orifice. 

t = operating time in seconds. 

a = area of the orifice in square feet. 

m = the coefficient for contraction. (See Fig. 299 ) 

G — gallon of 231 cubic inches discharged in the time t. 

V = velocity through the orifice in feet per second. 

Example 1. Fig. 284. How many gallons of water will be discharged in five min¬ 
utes, through an orifice of 0‘025 square feet, applied at 8 feet under the level of 
the water ? 

G = 37-75 a t = 37-75X0-025X5X60 ^8 = 800 gallons. 

Fig. 285. The weight P can represent the weight of a column of water whose 
P h' 

height = = q. 967 ; acting on the area A. 

Fig. 286. n — number of down strokes per minute, s = stroke of piston ; the 
air vessel C— 6 A s at the pressure of the atmosphere. 

Example 2. Fig. 286. How many double strokes must be made per minute by 
the lever of a hand pump, to throw up 22 cubic feet of water 18 feet high, in the 
time of 8 minutes and 15 seconds; the levers L = 30 inches, c = 8 inches, 
s = 0-6 feet, F = 20 pounds ? 8X60+15 = 495 seconds. 


3630 Q h’ e 3630X22X18X8 
t s FI ~ 495X0-6X20X80 


64-5 strokes per minute. 


Example 3. Fig. 294. A vessel of rectangular form is of dimensions A = 6 
square feet, the height h = 5 feet. What time will it take the water level to 
sink 2 feet, when the orifice a — 0"212 square feet. 


t _ A (ft — h!) _ 6(5 — 3) 

~ 2-52a(y/7i+y7i/ ~ 2-52X0-212(]/5+>/3) 


Motion of Water in Fipes^ 

Letters denote. 

L — extreme length of the pipe in feet. 

d — inside diameter in feet, and uniform throughout the length L. 


Example 4. Fig. 287. What will be the velocity of the water through a pipe of 
0-45 feet inside diameter, and L — 68 feet long, the head pressure of water being 
h — 8 feet ? 


r=48 



0-45X8 
68 +50X0-45 


9-6 feet per second. 









































































208 


Hydraulics, 



292. 

Q = 5-35 m b t(h\/ h — h' \Th% 
G = 40m b Z(A V /T— h'Jl?), 


5-35 rn b(h\/h — h' V h !) 


293. 

t = 0-95 /7 A( y/ h — JV ) 
f n / 7 717 ~ * 

A area of the vessel in square feet. 
t time in seconds, in which the water level 
will sink the space h — h'. 


Q — k b t. See Table for Weirs. 

t-S, b = R 

kb y kt y 


291. 

Q = 5-35 m b h t\f~h, 
G = 40 m b h ty/ 

_ _ Q _ 

5‘35 m b hV h’ 


294. 


A(h — h!) 


Am a (y/ h+ y/ h'), 


Weirs. 


290. 


7 i 


__Y_ 


Q = 4m a t( V h+ h'), 


295. 


3'85a rn 


(Jh - s/T), 


A y A A \f ft 

3*85£' rn 3*85 am 

























































Hydraulics. 


209 



18* 




































































210 


Hydraulics.—Hydrodynamics. 


Example 5. Pig. 289. Required the velocity and quantity of water discharged in 
a long pipe or hose of L = 135 feet lo' g, and d — 0-17 feet, attached to a hand- 
pump of D = 0-2 feet in diameter P = 44 pounds, and the end of the pipe ele¬ 
vated h = 20 feet above the piston Dl 


6 - 86 . / °' 17( ' 
V °' : 


(44 — 49X0-2^X20) 


1-95 feet per second. 


(•2(135+50X0-17) 

Q = l-95X5*38X0-2» — 0-042 per secondX60 = 2-52 cubic feet per minute, 
s = 0‘8 feet the stroke of piston, we shall have 

2-52 


n = 


0-8X0‘785X0-2 a 


= 100 strokes per minute. 


Table for Water flowing over Weirs. 


This Table is set up from careful experiments 
on a large scale, and is suited for weirs only. 
See Pig. 290. 

RULE. Multiply the width b in feet, of the 
weir by the coefficient k, and the product is the 
quantity of water discharged per second, in cubic 
feet, h is the height as represented by Pig. 290. 
The width b should be b ]> h. 

Example 6. How much water will flow over a 
wier of b = 5 feet, h = 0 5 feet in one minute ? 

Q = kbt = 1-1295X5X60 = 338 35 cubic feet. 


h. inches. 

0-4 

0-8 

1-2 

1-6 

h. feet. 

0-033 

0-066 

0-100 

0133 

m. 

0-424 

0-417 

0-412 

0-407 

k. 

0-01365 

0-05452 

0-10592 

0-16616 

2-4 

0-200 

0-401 

0-29171 

3-2 

0-266 

0-397 

0-44480 

4- 

0-333 

0-395 

0-63111 

6- 

0-500 

0-393 

1-1295 

8- 

0-666 

0-390 

1-7464 

9- 

0-750 

0-385 

2-0331 

12 

1-000 

0-376 

3-1350 


-»4- 


HYDRODYNAMICS. 

Water Power. 

The natural effect concentrated in a fall of water, is equal to the weight of the 
quantity of water passed through per second multiplied by the vertical space it 
falls. 

Fig. 297. Let Q be the quantity of water which passes through the orifice a in 
the time t = 1" second, in cubic feet of 62 - 5 pounds each. 

h = the vertical space the water Mis; then the value or natural effect of the 
fall is at the orifice a. 

P — 62-5 Q h, effects. 

But, Q = 5-06ap7T, then we have 

P=315-5a7tyC 

This will be in horse-power, 

1 w / 77^ 

H = 0-573 a h^h, h = 



n = 01134 Q h, 


h = 


H 


0-1134 £ 


Example 1. In a creek passes 18 cubic feet of water per second. How high 
must that creek be dammed up to produce an effect of 10 horses ? 


7i = 


10 


0-1134X18 


= 4-9 feet, the answer. 















W VTKU-VV IIICKI S. 


211 


WATER-W1I EELS. 

Water-wheels are of two essential kinds, namely, Vertical and Horizontal. 

The Vertical are subdivided into 

Overshot-wheels, Undershot-wheels, Breast-wheels, and High-breast aud Low-breast 
wheels. 

The Horizontal are with Floats, Screw-wheels, Turbin, Reaction-wheels, ilk. 

Waterwheels do not transmit in full the natural effect concentrated in a fall 
of water; under most favourable circumstances 80 per cent, has been utilized, 
but uuder poor arrangements only 20 per cent, may be expected. 

Example 1. Fig. 302. The vertical section of the immersed floats of an under¬ 
shot-wheel in a mid-stream is a = 27 square feet, velocity of the stream F= 8 - 6, 
and o — 4 feet per second, Required the horse-power of the wheel H — ? 

11 = —v) 3 =* ^^(8-6 — 4)> = 114 horses. 

200 v ' 200 v ' 


Example 2. Fig. 307. On a breast-wheel is acting Q — 88 cubic feet of water 
per second, the head li = 8 feet, velocity of the wheel at the centre of the 
buckets u=5 feet per second ; the water strikes the buckets at an angle u = 8° 
aud velocity V — 7 feet per second, ltcquired k the horse-power of the wheel, 
H = 1 


U 


A 8 

ii 


^( 8+ ^ ( - 7Xcos ' 8 °' _ 5 ' 1 ) 


= 65 horses. 


Example 3. Required the effect of Poncelet’s wheel, the head h — 4 feet, and 
the orilii e a — 5 square feet, the velocity of the wheel at the centre of pressure 
of the floats is v = 6‘78 feet per second ? 


V = 6 91 ] 4 13 82 feet per second. 

Q = o- 5X5X|/4 = 65 cubic feet per second. 

II = (13-82 — 6*78) =. 15-8 horses. 


Example 4. Fig. 309. A saw-mill wheel is to be built under a fall of h = 18 
feet, find to make n — 110 revolutions per minute. Required the proper diam¬ 
eter of the wheel. 


D = — yl8 = 3-857 feet, 

at the centre of pressure of the buckets. 

Felocity F = 8|/18 = 33-94 feet per second. 

Velocity v ==-^- = 22-2 feet per second. 

T he fall discharged 30 cubic feet of water per second. Required the horse¬ 
power of the wheel. H — l 

/f ^ 30 ^ 22^2 f33 . 94 _ 22 . 2 ) = 39 horses . 


IIow many square feet of dry Pine can it saw per hour ? 
See page 150. 30X39 ='1170square feet. 

The saw is meant to be applied dire t on the wheel shaft. 
















212 


Hydraulics. 



'302. 

Undershot wheel in a mid-sir earn. 

200 v J ' 

When V — 2v about, the effect will be, 
H= |L^, a = area of float. 


Undershot- Wheel. 


Poncelet's Wheel. 


228 


( V — v), when ft > 6 feet, 


Q v ( 

197' 


H ==- ^_( V — v) when ft < 5 feet, 


= 8ma^% V=WW1T 


Breast-Wheel with Parabolic drain. 


86-8 


’(r- t) v X 


When F = 2u, about, // = 

’ 0-47 























Hydraulics. 


213 



306. Low-breast Wheel. 

H= + ~S2~^ r °° su ~ ® )] 


Q = kb. V = -S. * See table for 1 



307. Breast Wheel. 


sr< F frH 



308. Over-shot Wheel. 


B- Si[ h + m (Vcosu - v) \ 


,y 7 , 7 . 35 7) +100, 

Proper velocity about n = ——- 

revolutions per minute. 


309. Saw-Mill Wheel. 

Proper diameter of the Wheel , 

100 /T - . fooi 

n = - y/ h , /««*> 

n 

n = revolutions per min. 


































214 


Atmosphere. Aerostatic. 


ATMOSPHERE. AEROSTATIC. 

The atmosphere round our earth, as well as all other gaseous matters, en¬ 
deavours to occupy a larger space to infinity, (no known limits,) but as it is a 
material substance, it is under the action of force of gravity, and cannot expand 
farther than when its density is in equilibrium with the said force. Conceive 
the atmosphere to consist of a great number of layers, one on the top of the 
other; the density of the under layers will evidently be greatest, because the 
upper ones press on them, and they are all elastic; hence the density of the 
atmosphere is greater at the surface of the earth than higher up. We can now 
find out the weight and density of all these layers. 


310. 


a 


A 


k 


a! 


A is a vessel full of mercury, in which is placed verti¬ 
cally a glass tube about 3 feet high above the surface l; 
in the glass tube is fitted, air-tight, piston a, just one 
square inch area, which can be moved by the piston-rod 
c ; now the piston stand is at a on the level l, and in con¬ 
tact with the mercury in the tube; raise the piston by 
the piston rod and handle c, the mercury in the tube will 
follow until the height of 30 inches, the piston still con¬ 
tinues to move higher in the tube, but the mercury will 
maintain its position at 30 inches from l. Now it may be 
supposed that it is some force of the piston that draws the 
mercury up in the tube; if so why did it separate at 30 
inches ? If the column beeomes too heavy it copld sepa¬ 
rate at l, and the 30 inches of mercury follow the piston; 
as this is not the case, but the weight of the atmosphere 
pressing on the surface l and forcing the mercury up in 
the tube until it (the mercury and the atmosphere) comes 
in equilibrium, which occurs at the 30 inches; and the 
piston only served to remove the atmospheric pressure 
in the tube; hence we have the weight of *a column of 
atmospheric air with one square inch base equal to the 
weight of a column of mercury 30 inches high and one 
sq. in. base. One cubic inch of mercury at 60° Fahr. weighs 0-491 pounds, this 
multiplied by the height, 30 inches, gives 14-73 pounds, the weight of the col¬ 
umns of mercury or atmosphere; this is generally termed “ the atmospheric 
pressure per square inch.” 

The specific gravity of mercury at 60° Fahr. isl3-58, and 




13-58X30 

12 


= 33 95 feet, the height of a coluinn of 


water required to balance the atmosphere. 

If the temperature and force of gravity were uniform throughout the atmos¬ 
phere, the density would decrease in.an arithmetical progression by the height 
from the ground, and by observing the altitude,of columns of mercury at two 
different heights, the extreme height of the atmosphere would be found simply 
by the formula 7, page 64, in which 

a — o the altitude of the column of mercury at the top of the atmosphere. 

b = 30 inches, the column of mercury at the level of the sea. 

8 = the difference of the columns of mercury at the level of the sea, and at a 
height h above the sea. 

Then, n multiplied by the height h, should be the extreme height of the 
atmosphere, or 


H-h 


(H 


i, 


s 





























Barometer. 


215 


Example. The mountain Chimborazo, Eucador, (South America,) is h = 3-87 
miles high above the level of the sea; at its top the column of mercury is ob¬ 
served to be only 27"63 inches, and 8 = 30 — 27 63 = 2-37. 


11 = 3*87 


(•2¥7 +1 ) 


+ 1 ) = 52'825 miles, the extreme height 


of the atmosphere. 

This is about the true height, but the calculation is incomplete by lack of 
many circumstances accompanied with higher calcules in mathematics, which 
can not be allowed to occupy room in this work. 

The column is 30 inches when the temperature of the atmosphere is 32° Fah. 
(See Barometer.) 




31L BAROMETER. 

The Barometer is based upon the same principle as the preceding experi¬ 
ment. It consists of a glass tube b, about 35 inches high, open at one 
end, c, and filled with distilled mercury ; inverted in a small vessel A. 
also containing mercury. About the top of the column of mercury is 
placed a scale to indicate the height from l. When disturbances take 
place in the atmosphere by heat, condensation. &c., its weight and den¬ 
sity will differ, and the column of mercury will fall and rise accordingly ; 
hence by connection of the scale at a, it indicates the disturbance. 


a- 


/ 



To Find the Density of the Atmosphere about a 

Barometer^ 

Jj Letters denote. 

h — altitude of the column of mercury in inches. 
t = temperature of the atmosphere, Fah. 

S specific gravity of the air around the Barometer at the time h and 1 
t are observed. 

S= I when t = 32°. 


S = 


h 


2 , 


0-0624(448-77 +t) 

Example. The Barometer has fallen to 26" 31 inches and the tem¬ 
perature is t = 60°. Required the specific gravity of the air t 

26*31 


S = 


= 0-83. 


0-0624(448-77+60) 

This formula does not include the expansion of mercury from 32° to 60°, 
which must be separately reduced by the following formula. 

h = H (0-9967962 — 0-000100K) 3, 

in which H = the observed column at the temperature t, and h — the true 
col umn to be inserted in the formula 1. 

To Measure Vertical Heights by the Barometer. 

Letters Denote , 

H = column of mercury | t the Iower Ktation . 

T = temperature of the air j 

h — column of mercury ' at the higher station. 

t = temperature of the air j ° 

l = latitude of the place. 

f = vertical height, in feet, between the higher and lower station. 


L 




















216 


Wind. Areodynamio. 


X ~ l0g- ( 7i X 1+0-0001001(T— t) 

/= 60345-5Lr(l+0-002551 cos.2l)(l+0-00208(T-H — 64°)), - - 5. 

If the atmosphere is yery calm the observations may he made one after the 
other by one Barometer and detached Thermometer; but the least disturbance 
of wind requires the observations at the upper and lower stations to be made 
at the same time. The reduction of the columns of mercury is included in the 
formula 5. 



WIND. AREODYNAMIC. 

The motions and effects of gases by the force of gravity, are precisely the same 
as that of liquids. (See Hydraulics.) 

The altitude or head of the atmosphere at uniform density will be the alti¬ 
tude of a column of water 33-95 feet, divided by the specific gravity of the air, 
0-0012046, or, 


33-95 

0-0012046 


= 28183 feet, 


the velocity due at the foot of this head is (Formula 1, page 183.) 

F = 8 02 j/28183 = 1346-4 feet per second, the velocity at which the air will 
pass into a vacuum. 

Velocity of Wind. 

When air passes into an air of less density, the velocity of its passage is mea¬ 
sured by the difference of their density. 

^ ^ | density of the air in inches of mercury. 

t = temperature at the time of passage. 

V = velocity of the wind in feet per second. 


F = 1346-4 



1+0-00208^, 


6 . 


The force of wind increases as the square of its velocity, 
a = area exposed at right-angles to the wind, in square feet. 

F= force of the wind in pounds. 

It = horse-power. 

v = velocity of the plane a in direction of the wind, -f when it moves oppo¬ 
site, and — when it moves with the wind. 

F = 0-002288a F*, 

F = 0-002288a(Fqrr)!», 
ar(F ~a) 3 


when v = o, — 


7 , 

8 , 


H = 


y. 


241400 

Example. A Rail-train running ENE 25 miles per hour, exposes a surface of 
1000 square feet to a pleasant brisk gale NE by E. Required the resistance to 
the train in the direction it moves, and the horse-power lost ? 

ENE—-NEby JST= 3 points = 33° 45'. 

F = 14 feet per second, a brisk gale. 
v = 25Xl - 467 = 36-6 feet per second. 

F= 0-002288 sin.^33° 46'Xl000(14+cos.33° 45'X36’6)» = 3051 pounds. 


II — 


3051X36-6 

550 


20 horses. 


550 

















Table of Velocity and Force of Wind.—Balloon. 


21 ? 


M les 

feel per 

Force per 

per hoar. 

second. 

sq.ft pound. 

1 

1*47 

0-005 

2 

2-93 

0-020 

3 

4-4 

0-044 

4 - 

5-87 

0-079 

5 

733 

0-123 

6 

8-8 

0-177 

7 

10-25 

0-241 

8 

11-75 

0-315 

9 

13-2 

0-400 

10 

14-67 

0-492 

12 

17-6 

0 708 

14 

20-5 

0-964 

15 

22-00 

1-107 

16 

23-45 

i-25 

18 

26-4 

1-55 

20 

29-34 

1-968 

25 

36-67 

. 3-075 

30 

44-01 

4-429 

35 

5134 

6-027 

40 

58-68 

7-873 

45 

66-01 

9-963 

50 

73-35 

12-30 

55. 

8%7 

14*9 

60 

88-02 

17-71 

65 

95-4 

20-85 

70 

102-5 

24-1 

75 

HO- 

27-7 

80 

117-36 

31-49 

100 

146-66 

50- 


Common Appellation of One, Force, of 
Wind. 

Hardly perceptible. 


r 


J ust perceptible. 


Gentle pleasant wind. 


Pleasant brisk gala. 


Very brisk.. 

| High; wind. 

Very high. 

Storm or tempest. 

Great storm. 

| Hurricane. 

Tornado, tearing up trees, Ac. 


BALLOON. 

To Find what Weighty and to what Height a Balloon can 

raise* 

Letters denote. 

C = cubfc contents of the balloon, in feet. 

s = specific gravity of the gas used to inflate the balloon, air = 1 at 32°. 

W = the weight in pounds, it can raise from the ground. 
w = tlie weight with which it fs loaded, including the weight of. the materials 
of which it is made. 

f = height in feet to which k will raise. 

T= temperature at the ground. 
t = temperature at the height/. 

H — Barometer column in inches, at the ground 1 . 

I = latitude of the place. 

W — 0*07529(7(1 — s), ...... W, 

_ i t Hw w_ 1 

S _ JOg.^ w X 1+0 . 0001()01 ^y_ J) 

f= 60345-5t*(l-ffi-002551 ces.2Z)(T+0-00208(F+ t — (A)) • 12. 

Balloons are commonly filled with Hydrogen gas whose specific gravity is 
» — 0‘07, when pure, or about 14 times lighter than air. of which say 10 times to 
be relied upon, as some foreign heavier gases- may accompany it.. 



19 











Wiwd-Mit,ls. 


218 


WIND-MILLS. 


The sail-shaft of vertical wind-mills should have an inclination from 12° to 
15° with the level when built on low flat grouud ; on high ground, elevated 
from 1000 to 1500 feet within a circle of about two miles, the sail-shaft should 
incline from 3° to6 b with the level 

Effect of Wind-Mills. 

Letters denote. 

A = projecting area of sails exposed to the wind, in square feet. 

V = velocity of the wind in feet per second. 

H = horse-power of the mill. 

R r = hmer me } radH of 8ails in feet ‘ 

l = n/ radius of centre of percussion in feet. 

A 

n = number of revolutions of sails per minute. 

v = mean angle of sails to the plane of motion. 

The angle of the sails should be from 20° to 30° at the inner radius r, at the 
extreme radius R from 7° to 12°, and the meau angle v — 15° to 17°. 

Ain sin.v cos.v / y _ bn sin.v \ 3 

\ ) 


H= 


1,540,000 \. 9-5 

assume the mean angle v = 16°, we have the horse power. 

Ain f 
5,800,000\ 

In order to utilise the maximium effect of wind, it is necessary to load the 
mill that the number of revolutions of the sails are proportional to the velocity 
of ihe wind. 

Proper revolutions will be found by n ' 11 ^ 


Ifw =-16°, n = 


11-6 V 


27=- 


A Fa 


l sin. v 

and A = 1 - 185 -? 0 ° .g. 

V s 


? 


212 75 = 6 feet wide by 


l 1,135,000 

Example 1. A wind-mill is to be built of six horse power in brisk wind, 
V = 20 feet per second. Required the area of sails A 

A =M 3 ?-, n() ° x6 . = 851 sq. feet. 

203 
851 

Example 2. Four sails — = 212-75 sq. feet each. 

35 - 5 long, dimensions of the sails. Inner radius r = 5 feet and R = 5-f 35'3 
=40-5 feet. Required the radius of centre of percussion l — l 

1= V 40 '5 ,J + 5 !> = ./ 832-tT=28-85 feet. 

2 

Example 3. The mean angle of sails to be v = 16°. Required the proper 
number of revolutions of the sails per minute in brisk wind ol F=20 feet per 
second n, — Z 

Revolutions » = ’ >X 2 2.= 8 per minute. 

28-85 

Example 4. A wind-mill has an area of A = 750 sq. feet exposed to high wind 
of V — 50 feet per second, and makes n = 26 revolutions per minute,—centre 
of percussion l = 25. Required the horse power of the mill H = ? and proper 
number of revolutions per minute n = ? 

_ 750X26X26 f . Q 25X27 ^ 


5,800,000 
„_11'5X50_ 
26 

rr _ 750X 25X23 f m 
5,800,000 V 


34-5 

23 rev. per minute. 

25X23 


7'8 horses. 


34-5 


8-26 horses. 




























Light.—S oujm 


219 


Composition of air by weight, 


(75-55 
■1 22-32 
( 1-32 


7 5*55 Nitrogen. 

Oxygen. 

Carbonic acid and water vapor. 


Composition of water by weight, j Hydrogen. 


Composition of water by bulk. 


f 1 Oxygen. 

\ 2 Hydrogen. 


Height of columns that press one f || 3 feet « of Salter ^ 

pound per square inch. inches ^f merely at 60-°. 


LIGHT, 

Light is the sensation transmitted by the eye and produces the sense of 

seeing. 

Heat and Electricity produce light by making bodies luminous. 

Intensity of Light is inverse as the distance from the luminous body. 

Velocity of Light is 192500 miles per second. 

Light passes from the sun 95000000 miles in 8 minutes. 

Light can pass around the whole earth in £ of a second. 

Solids must be heated to at least 600° to produce light in the dark; and to 
1000° in day-light. 


-♦ 4 - 


S 0 U N D . 

Velocity of Sound through Air* 

v = velocity in feet per second. 
t = temperature of the air, Fah. scale. 

D — distance in feet the sound travels ;in the time T. 


v = 1089-42 yi+0-00208(t —32), 

Velocity of sound in water is about 4 times that ;in air, and 8 times that 
through solids. 

Intensity of sound is inversely as the square of the distance, 

D = 1089-42T j/1+0 0020S(T—32), 

T— £ 

v * 

Example. A ship at sea was seen to fire a cannon, and 6*5 seconds afterwards 
the report was heard, the temperature in the air was 60°. Required the dis¬ 
tance to the ship ? 

D = 1089-42X6-5 1/1+0-0028(60° — 32) = 7300 feet, or 1-38 miles. 


Descriptions of Sound. 

A powerful human voice in the open air, no wind, 

Report of a musket,. 

Drum,. 

Music, strong brass band, ..... 

Cannonading, very strong,. 

In a barely observable breeze a strong h urn an voice 
with the wind can be heard, 


Audible at a distance of 


feet. 

460 

16000 

10560 

15840 

575000 

15840 


miles. 

0*087 

3*02 

2 

3 

90 
























220 


Bells. 


RINGING BELLS. 

Letters denote. 

D = diitmeter of the bell at the mouth, in inches. 

d — diameter of the bell at the crown, in inches. 

h = heighth of the bell from the mouth to the crown in inches. 

S = thickness of sound bow in inches. 

W— weight of the bell in pounds avoirdupois. 

n — number of vibrations per second, corresponding with the key note of the 
boll, and to be found iu the accompanying table I. 

Jc = from 0'07 to 0-08, or a coefficient expressing the relative thickness of the 
sound bow to the diameter of the bell. In peals of bells, the sound bow is 
generally S = 008D for the triple, and S = 0-07Z> for the tenor; the interme¬ 
diate bells in the peal having the intermediate proportions of sound bow. 

Example 1. Required the weight of a bell D = 62 inches in diameter, and S 
= 4£in. thickness of the sound bow, W = ? 

Formulae 1. TF= 0-25X62*X4’5 = 4324-5 pounds. 

Example 2. A bell of 2,500 pounds is to be constructed with a sharp note, 
taking the sound bow k == 0-075. Required the diameter of the bell Z> = ? 

Formulae 10. D = >/ = 51-084 inches 

0 0i5 

Example 3. It is required to construct a bell with the key note in the 
first octave above zero, n =152-22. To be of light weight with a full good note, 
for which latter case take k = 0'07. Required the diameter of the bell, D = ? 

Formulae 11. D — == 26 665 inches. 

Ia2'22 

Example 4. Required the key note of a bell with D —■ 36 5 in. diameter and 
S — 2-75 in., n = ? 

2"75 

Formulae 4. » = 58000 X og m — 119-7 vibrations. 

36.52 

In the table the nearest number 120 82, in the first octave below zero, answers 
to the key note B, which will be the note of the bell. 


Example 5. A bell of 6860 pounds is to be constructed with the key note Ct 
in the first octave below nero n = 64, see table I. Required the diameter of 

the bell D = ? _ 

, >6860 

Formulae 9. D = 21-947. / -xr- = 70*6175 inches 

64 


4 /o 

h D = 21-947. / 


Example 6. Required the thickness of sound bow for the bell in the pre¬ 
ceding example? D = 70.6175 inches and « = 64. S = ? 

Formulae 12. 5 = = 5-5027 inches. 

O 80 UU 

Example 7. Required the weight of a bell D — 48 inches diameter at the 
mouth, d = 25 inches at the crown, and h = 34 inches height from the mouth 
to the crown, S = 3*5 in., W = ? 

Formulae 17. 

Tf'= 48X25X3*5 (0*5—0-002816X25) +0-00375X34X 252x 3-5 = 2126*226 pounds. 



















Bells. 




221 


Formulas for Ringing Bells® 


W= 0-25 D*S 

nr_ D*n__ 

232000 * 

W = 0-2bD3Jc - 


n = 58000-^ 
i> 9 


n = 232000 J- 


n =» 58000^ - 


W 

S 


v 

v'? 

4 / if 

'V ^ 

3 /r^ 

‘ / X 


D = 240-83 

Z> = 21-947 


IF 


V 


D — 58000' 


7 

8 

- 9 

- 10 
- 11 


S = n P* 
58000 


S = 


4 W 


Z?» 
S = k D 


k - S 


k — 


4 W 

~Efl 


12 

13 

14 

15 

16 


W '-= Dd £(0-5—0-0002816 d) +0-00375 h d» S .... 17 


Table I. Vibrations per Second — n. 


Key 

note. 

3rd Oct. 

Bass. 

2nd Oct. 

1 

1st Oct. 6 

6 

l 1st Oct. 

Descant. 

2nd Oct. 


C 

16-000 

32-000 

64-000 

128-00 

256-00 

512-00 

C# 

16-947 

33-385 

67-790 

135-58 

27100 

542-32 


17-960 

35-920 

71-840 

143-68 

287-36 

574-72 

D i 

1*9-027 

38-055 

76-110 

152-22 

304-44 

608-88 

E* 

20-159 

40-318 

80-636 -1 

161-27 

322-54 

645-09 

F 

21-357 

42-715 

85-430 

170-86 

341-72 

683-44 

F i 

22-627 

45-255 

90-510 

181-02 

362-04 

724-08 

G* 

23-972 

47-945 

95-890 

191-78 

383-56 

767-12 


25-398 

50-797 

101-59 

203-19 

406-37 

812-75 

A* 

26-908 

53-817 

107-63 

215*27 

430-53 

861-07 

A i 

28-508 

57-017 

114-03 

228-07 

456-13 

912-27 

B 

30-204 

60-409 

120-82 

241-63 

483-27 

966-54 

C 

32-000 

64-000 

128-00 

256-00 

512-00 

10240 


Table V. 


Abscissa 

X 

Ordinate 

y 

S=1 

Thickness 

S = 0-07 D 

of Metal. 

8 = 0-75D 

S = 0*081? 

1 

0-4142 

1 , 

0-700 

0*750 

0-800 

H 

0-686 

0-800 

0-560 

0-600 

0.640 


0-867 

0-653 

0-459 

0-490 

0-522 

2* 

0-974 

0-547 

0-382 

0-410 

0-43J 

3 

1025 

0-474 

0-331 

0-355 

0-379 

H 

1-030 

0-423 

0-295 

0-317 

0-338 

4 

1-000 

0-380 

0-266 

0-285 

0 304 


0-955 

0-351 

0-245 

0-263 

0-281 

5 

0-875 

0-327 

0.228 

0-245 

0-261 

H 

0-775 

0-301 

0-211 

0-226 

0 241 

6 

0"665 

0-291 

0-203 

0-218 

0-233 

6* 

0-530 

0-286 

0-200 

0-214 

0-228 

7 

0-390 

0.279 

0195 

0-209 

0 223 

n 

0-235 

0.272 

0-190 

0-204 

0 217 

8 

00"5 

0.267 

0-186 

0-200' 

0-213 

8-74 

0-78 

0.333 

0.233 

0.250 

0-266 





























































Bells. 


--f 


When a bell is to be constructed, we generally have the weight or key note 
given by contract, the diameter and sound bow are calculated by the preced- 1 
ing formulaes and examples, and then ready to proceed with the construction. 

The diameter of the bell at the mouth, is divided into 10 equal parts, called ; 

• strokes, which then is the scale and measurement for the construction. Make 1 
a decimal scale, as shown on plate VII. 

The section of a bell is generally laid out on a piece of board represented by 
the dotted lines a, b, c, d. which then is cut out and used for turning up the 
mould for the boll. The board should be about 11 strokes long, and 2 5 strokes : 

! wide. Through the centre of the board draw the line p, q , parallel to b, c, _ 

I bisect the line p, q, and set four (4) strokes from the bisecting point towards 
each end, divide the strokes into halves, and number them as shpwn on the t 
accompanying drawing. Through each division draw lines at right angles to I 
i p. q, set off the corresponding ordinates y expressed in strokes, table II, and j 
i join them by a curve-line, which then will be the centre of thickness of metal \ 
| in the bell. ; 

In the end of tlje first ordinate, as a ceptre, d r & w a circle with a diameter 
i equal to the desired thickness of the sound bow, which should be from 0 7 to 
0 8 strokes. At every succeeding ordinate draw a circle with the diameter ] 
noted in table II; for instance, if the thickness of the sound bow is I 
inches, then the thickness of metal or diameter is the circle at the third 
ordinate will be 4-5X0*174 = 2 133 inches ; but if the sound bow is 07,0*75 or 0.8 I 
strokes, the thickness of metal at the third ordinate will be 0-331, 0-355, or 0.379 
strokes. When all the thicknesses are thus drawn, draw the two lines tan- 
genting the circles on each side of the centre line of the metal. From the 
orige o, draw the tangents to the sound bow; the lino o*l should not quite 
tangent, but leave a moulding about 0*02 strokes, as represented by the 
drawing. Prolong the ordinate, and set off 179 strokes to e. whiph then is 
the centre for the curve on the top, draw the arc through the centre of the 
small circle at the 8th ordinate; join e, 8, set off from e, 0-40 strokes to the 
centre for the inside curve at the top. 

Thickness of metal of the top should be 0*3 the sound bow at 8, and 0-333 j 
at r. 

Draw the ordinate at 8-74, set off 078 to r, join r and the abscissa 8-48, and 
prolong the line through r; then finish the drawing as shown on the plate. 

When the board is cut out and ready for turning the mould, it must be 
carefully set, so that the outside diameter of the crpwn will be half the diame- j 
ter of the mouth of the bell. 

Should the constructor desire to make the bell lower in proportion to the 
diameter, he can do so by shortening the radius for the top of the crown, ; 
or lower ng the crown ; though not lower than to the seventh ordinate, which 
only affects the continuation of the sound. But whatever height chosen, it is 
necessary to maintain the diameter of the abscissa at 7*9 strokes, equal to half i 
the diameter of the mouth of the bell, or to maintain the same position of 
the hqard tQ the axis of the bell, in order to attain the mapimum quality of 
tons. 

Clapper* 

The weight of the clapper should be from one-fortieth to one-fiftieth the 
weight of the bell, the small bells take the largest clapper, and vice versa. 


Bell Metal. 

Thirty pounds of Tip to Qne hundred pounds of Copper is a good proportion. 


222 


To Construct a Bell* 




















Plate, YU 


w 




\ 

v 

\ 

\ 


\ 


























































































r ,?P 

r -t*. -•'■ 'tW T ~* .'ffi. • --P 





















■X 





..... -i,- Vv _ . . 


' • , 

•—< —■ 















■ . 












■ ' ; '■ ' : ■ 


■Hi. 












t •' ' 








. 























4* 

















■ > V' • 


■ -■ 1 












V 







' 



. 












































Music. 


223 



Diatonic Scale in Descant Clef* 
































































































































224 

— 


Heat. Caloric. 


HEAT. CALORIC. 


The Physical constitution of heat is yet under investigation by operative minds, 
its well known character and effect upon matter is the base for the investiga¬ 
tion. 

Heat resembles light, electricity and magnetism, and is thus far assumed to be a 
material substance. 

Heat is contained in all matters, with no known exception. Tw i o bodies con¬ 
taining different quantities of heat per unit, being placed in contact,—the heat 
will pass from,one to the other until it comes in equilibrium, that is when 
the two bodies contain equal quantities of heat per unit. It is this passing of 
heat that first comes under our notice. The body from which the heat passes 
will feel the other to be cold, and vice versa, —the one that receives the heat will 
feel the other to be warm, until there is no further passage, namely, when the 
bodies will feel neither warm nor cold tp each other; hence the measure of the 
emplix sensible heat, is the difference between the heat per unit in the two 
bodies. Cold is only a want of heat. 

Caloric is only another word of expression for heat. 

Caloric is of two kinds, sensible and latent. 

Sensible Caloric is that which is sensible to the touch, felt as cempera- 
ture, and can pass freely from one body to another. 

Latent Caloric is that which is insensible to the touch. It is contained 
in bodies without being felt as temperature, but can by chemical action become 
sensible ; for instancy a piece of burned limestone put into water will get warm 
and heat it, although both were cold before ; the latent caloric in the water was 
set free to sensible. 

Influence of Heat, on Matter’s Coherence. 

All bodies in nature expand when heated, and contract when cooled. Solid 
bodies vary but little by the difference in temperature. Liquids vary more, but 
gases are extremely susceptible to the impression of heat and cold. 

Table of Linear Expansion of Solids. 


Difference in 

Length = 1 at 32°. 

Length at TP 

k difference length 

temperatures. 

Names of bodies. 

per degree. 

32° to 212° 

) ( 

1-00086183 

0-00000478 

32 to 392 

> Glass, - - - J 

1-00184520 

0-00000546 

32 to 572 

i ( 

1^00303252 

0-00000583 

32 to 212 

32 to 572 

| Wrought iron, " * { 

1-00118210 

1-00440528' 

0-00000656 

0-00000894 

32 to 212 

Soft good iron, 

1-00122045 

0-00000680 

32 to 212 

Iron wire, - 

1-00123504 

0-00000687 

32 to 212 

Cast iron, ... 

1-00111120 

0-00000618 

32 to 212 

Soft steel, * 

1-00107915 

0-00000600 

32 to 212 

Steel hardened and tern. 150°, 

1-00123956 

0-00000689 

32 to 212 

| Copper, { 

1-00171820 

0-00000955 

32 to 572 

1-00564972 

0-00001092 

32 to 212 

Lead, .... 

1-00284836 

0-00001580 

32 to 212 

Gold, pure, ... 

1-00146606 

0-00000815 

32 to 212 

Gold, hammered, - 

1-00149530 

0-00000830 

32 to 212 

Silver, pure, - 

1-00190868 

0-00001060 

32 to 212 

Silver, hammered, 

1-00201000 

0-00001116 

32 to 212 

Brass, common cast, 

Brass wire or sheet, 

1-00187821 

0-00001043 

32 to 212 

1-00193333 

0-00001075 

32 to 212 

32 to 572 

| Platinum, pure, - - j 

1-00088420 

1-00275482 

0-00000491 

0-00000520 

32 to 572 

Platinum, hammered, - 
Zinc, pure or cast, 

1-00095420 

0-00000530 

32 to 212 

1-00294167 

0-00001633 

32 to 212 

Zinc, hammered, - 

1-00310833 

0-00001722 

32 to 212 

Tin, hammered, - 

1-00270000 

0-00001500 

32 to 212 

Tin, cast, ... 

1-00217298 

0-00001207 

32 to 212 

Fire brick, ... 

1-00042280 

0-00000235 

33 to 212 

Marble, .... 

1-00110410 

0-00000613 

32 to 212 

Granite, .... 

1-00078940 

0-00000438 













Heat. Caloric. 


225 


Table of Volume Expansion of Liquids* 

Difference in 
temperatures. 
32° to 212° 
212 to 392 
392 to 572 

32 to 212 

32 to 212 

32 to 212 

32 to 212 

32 to 212 

32 to 212 

Names of Liquids. 

Mercury, 

U 

Water, - 

Salt, dissolved, - 

Sulphuric acid, - 

Oil of Turpentine and Ether, 

Oil, common, 

Alcohol and Nitric acid, 

Volume at TP 

1*00018018 

1*00018433 

1*00018868 

1*00046600 

1*00050000 

1*00060000 

1*00070000 

1*00080000 

1*00100000 

k difference in vol. 
per degree. 
0*000001000 
0*000001025 
0*000001048 
0*000002595 
0*000002778 
0*000003333 
0*000003890 
0*000004444 
0*000005555 


All gases expand and contract equally and uniformly; 0*0020825 its volume per 
degree of Fah. thermometer. The accompanying Table is the result of Mr. 
Dalton’s experiments with air. The volume at 32° is equal to 1 or the unit. 


Table for Volume Expansion of Air* 


Degrees. 

Volume. 

Degrees. 

Volume. 

Volume 

32° 

1*000 

80 

1*1110 

1 6? 

33 

1*002 

85 

1*121 

1 J 

34 

1*004 

90 

1*132 

W<f 

35 

1*007 

100 

1*152 

AW 

40 

1*021 

200 

1*354 

i i 

45 

1*032 

212 

1*376 

i i 

50 

1*043 

302 

1*558 

i 9 

55 

1*055 

392 

1*739 

1 3 

I IT 

60 

1*066 

482 

1*919 

1« 

65 

1*077 

572 

2*098 

2 sS 

70 

1*089 

680 

2*312 

2 A 

75 

1*099 


1 



Letters Denote, 

L — length or any linear measure of the body of the temperature T. 

I _ length or linear measure at the temperature t. 

V— volume, of liquids at the temperature T: 
v = voluxue at the temperature t. 

j c _ coefficient for the linear measure or volume as noted in the Tables. 
The volume of solids is as L 3 : I 3 . 

The linear measure of liquids is a v : v V. 

Formulas of Linear Expansion of Solids* 


L = 


< 


lll+k(T 



l = 


L 

l+k(T—t)’ 


T =irr +t ’ 


t = 


T 


L — l 
k l * 


Example 1. A copper rod of L = 22*55 feet long is 140° warm, 
perature must it be cooled to fit in a space of l = 22*52 feet ? 


To what tem- 


22*55 — 2*2*52 
22*52X0‘bfib0158 


= 55*7° the answer. 


J 


t = 140 































226 


Heat. Thermometers. 


Formulas of Volume Expansion of Liquids* 


F=tVi-b&(r—o), 


T= Z-—+t, 


v — 


k v 


t = T 


i+kcf-ty 

Y — v 


kv 


Example 2. A vessel containing 5’68 cubic feet of water at t — 42°, is closed 
up round the water, but a cylindrical pipe of 0-008 square feet, inside section, is 
raised up vertically from it; now let the temperature of the water be raised to 
T = 130°. How high will the water rise in the pipe ? 

F = 5-68 [1+0-000002595(130 — 42)] = 5-681297 cubic feet, 

, 5-681297 — 5-68 . , 

and ----= 0-162 feet = 1-945 inches, 

O’uUo 


the height to which the water will raise in the pipe. 

This is the principle upon which Thermometers are constructed, but the scale 
can only be approximated by this formula. The substances adopted for ther¬ 
mometers are spirits of wine and mercury; oil and ether has also been proposed, 
but the two former are best, and mercury is most generally -used. 

-»ji-.- 

, THERMOMETERS. 

There are three different graduated Thermometers in use, namely Fahren¬ 
heit’s Oddus's, and Beamur’s. 

The first one, or Fahrenheit’s is used in North America, England, and Holland. 
The second or Celcius's in France, Sweden and Germany. 

The third one, or Beamur’s, was formerly used in France and some parts ot 
Germany, but now only in Spain. 

The Figures exhibit their difference. 

Fa hr. Celci. Ream. 



Proportional Formulas for the 
Thermo metrical Scales* 

Celoi. — | Ream. = | (Fah. — 32.) 

Ream. = ^ Celci. = | (Fah. — 32.) 

Fahr. = f Celci,+32 = f Ream.+32. 

Example. How much is 68° Celcius on Fahren¬ 
heit’s scale. 

Fahrenheit’s = |x68 +32 = 154-4°, the answer. 


Fluid boils when its vapour has the same density as the atmosphere where it 
boils, hence, fluid will boil sooner high up in the atmosphere than at the 
ground. 

In vacuum water boils at 88°. 

The mean temperature of the earth is about 50°; at the torrid zone 75°' 
temoerate : oae 50°; and in the polar regions 36°. 

Water cm be kept in liquid to 20°. 






















Table or Tempebatoes. 


227 


Tuhle of Temperatures \v 

SMELTING POINTS. 

Cast iron, fully sin., - - 27549 

Gold, fine, .... 1983° 

Silver, fine, .... 1850° 

Cooper, . 2160° 

Brass, common, ... 1900° 

Zinc. ..... 74 o° 

Lead,. 594° 

Bismuth, - - - - 476° 

Tin,.421° 

1 Tin, 1 Bismuth, - - 283° 

3 Tin, 2 Lead, 5 Bismuth, - 212° 

1 Tin, 1 Lead, 4 Bismuth, - 201° 

Antimony, ... - 790° 

Sulphur, .... 228° 

Phosphorus. - 109° 

Beeswax, white, ... 155° 

“ yellow,... 142° 

Tallow, ----- 92° 

Ice,.32° 

Oil of Turpentine, - - 14°. 

Ice of strong Brandy, - 7° 

1 Snow and 1 Salt, - 0° 

Mercury, ... - — 390 

lien Bodies change Form* 

BOILING POINTS. 

Mercury, .... 630° 

Oil of Linseed, ... 6 ?) 0 ° 

Sweet Oil, .... 412 0 

Sulphuric acid, - - • 410° 

Sulphur, .... 390° 

Phosphorus, - • 374° 

Oil of turpentine, - - 315° 

Sea-water, salt, - • - 217° 

Water, distilled, - - - 212° 

Alcohol, - . . 174° 

MISCELLANEOUS. 

Metals, red, daylight, - - 1077° 

Iron red, daylight, - - 884° 

Common fire, ... 790 ° 

Iron bright red, in dark, - 752 0 

Human blood is - - - 98° 

Cold greatest ever produced, — 90° 

Venous fermentation, - — 60 to 70° 

Acetous fermentation begins, — 78° 
Acetification ends, - — 88 ° 

Phosphorous burns, - - — 43° 

A comfortable room about 60° to 70° 

Tal>le of Power for Transmission of Heat* V 

CONDUCTING POWER. 

Gold, - - - 1000 

Silver, ... 973 

Iron, ... 347 

Tin, - - - * 304 

Copper, ■ 898 

Zinc,.... 363 

Lead, ... 180 

Platinum, ... 981 

Marble. ... 24 

Fire-brick, - 11 

Fire-clay, ... 11*4 

Porcelain, * - - 122 

Water as the Standard. 

Water, - - - 10 

Pine, - -39 

Lime, ... 39 

Oak, .... 33 

Elm, ... 32 

Ash, - - - 31 

Apple, - - - 28 

Ebony, ... 22 

RELATIVE CONDUCTING POWERS 
OF SOLIDS. 

Hare’s fur, - 1*315 

Eider-down, - - J305 

Beaver’s fur, * * 1296 

Raw silk, ... 1284 

Wool, - - - Ml 8 

Lamp-black, - - 1*117 

Cotton, ... 1-046 

Lint, - - - 1-032 

Charcoal, - - - 0-936 

Ashes of wood, - - 0-927 

Sewing-silk, - - 0-917 

Air, - - - 0-577 

RELATIVE CONDUCTING POWER OF 
FLUIDS. 

Mercury, - „ - - 1000 

Water, - - - 357 

Proof spirit, * - 312 

Alcohol pure, • - 332 

RADIATING POWER. 

Water, ... 100 

Lampblack,... 100 

Paper, writing, - - 98 

Rosin, ... ge 

Sealing wax, • - 95 

Glass, common, - 90 

India ink, ... 33 

Ice, .... 85 

Red Lead, ... 39 

Graphit, - - - 75 

Lead, tempered, - - 45 

Mercury, ... 20 

Lead, polished, - - 19 

Iron, polished, - - 15 

Tin and Silver, - - J 2 

Copper and Gold, - - 12 

REFLECTING POWERS. 

Brass, - - - ]00 

Silv r, 90 

Tinfolium ... 35 

Tin, .... go 

Steel, ... 70 

Lead, ... 60 

Glass, ... 10 

Glass, oiled or waxed, - 5 

Lampblack, ... 0 































228 


Specific Caloric. 


-l 


Mixtures of. 

Nitrate of Ammonia, 
Water, 

Sulphate of Soda, 8 

Muriatic Acid, 5 

Dilute Sulphuric Acid, 5 ) 
Snow, 4 j 


Cold produce. 

!} 460 

j 50° 

23° 


Degrees Fa.hr. 
From+50° to +4°. 

From+50° to +0°. 
From —68° to —91°. 


-» 4 - 


SPECIFIC CALORIC. 

Specific Caloric is the relative quantity of heat contained in bodies of equal 
weight or volume, and of the same temperature. 

Let two different substances of known weight or volume and temperature, be 
mixed together; the temperature of the mixture will dissolve the relative 
quantity of caloric in the ingredients. 

Mixture of tlie same Substances* 

Fetters denote. 

W = weight or volume of a substance of temperature T. 
w — weight or volume of a similar substance but temperature t. 

V = temperature of the mixture W-\-w. We shall have, 


t'(Wfw) = WT\-wt, 


Jf W T+w t 
t - u/ , . "» 


W- 


w(t' — t) 


iV-j-w 
w(l' — t ) 


+ *. 


T—t! ’ W 

Example 1. Let W == 4-62 cubic feet of water at T — 150° be mixed with 
w = 5*43 cubic feet at t = 46. Required the temperature of the mixture t' = ? 


t' = 


4-62Xl50°4-5-43X46° 

4*62+5*43 


= 97*6° the answer. 


27*3 gallons 


Example 2. How much water of T = 107° must be mixed to w 
of t = 58°, the mixture of the water to be 75° ? 

w== g'y? 0 - 58 ) = 14*5 gallons. 

167 — 7o b 

Mixture of different Substances* 

TV and w expressed by weights only. S and s == Specific caloric as given in 
the accompanying Table. We shall have, 

— v) = w s(t' — t ), v ws T+w s 1 


W= w *( tf ~ ® 

w-ty 


W S^- w s 
t'( WS+w s) — w st 

1 ~ wa 


Example 3. To what temperature must W = 20 pounds of iron be heated to 
raise w — 131 pounds of water of t = 54° to a temperature t' — 64°. 1—1 

From the Table we have s = 1, and S = 01218. 

64(2 0X0*1218+1.31)— 131X1X54 
20X0-1218 


T== 


= 602 °. 


the required temperature, supposing no vapour escapes from the water. 

If any chemical action takes place in the mixture, these formulas will not 
answer, because part of the sensible caknic may become latent, or latent calorie 
may be set free. 













Table of Specific Caloric. 


229 


Table of Specific Caloric* Waler as Unit* 


Names of Substances. 


Water, - 
Iron, 

Glass-crystal, - 
Mercury, 

Lead, 

Tin, 

Sulphur, 

Lime, burned, - 
9 Water, 10 Lime, 
Sulphuric acid, sp. g 


Nitric acid, 

A lcohol, 
Platinum, 
Antimony, 

Zinc, 

Copper, - 
Iron, 

Glass, - 
Gold, - 
Bismuth, 

Woods in average, 
Sweet Oil, 

Nickel, 

Cobalt, - 
Tellurium, 


ep- g 
sp- g 


= P8/058, 
= 1-29896, 
= 0-81, 


SPECIFIC CALORIC OF GASES AT EQUAL 
DENSITY. 

Air, atmospheric, ... 
Hydrogen, - * - - 

Oxygen, .... 

Nitrogen, .... 

Carbouie-oxid gas, 

Carbonic acid, .... 
Nitro-oxid gas, - 

Gas of oils, .... 

Steam, * 


Specific Caloric. 


32° to 212°. 

1.0000 


32°, 572°. 


0.1105 

0.1929 

0.029 

0.02819 

0.04755 

0.2085 

0.2169 

0.43912 

0.3346 

0.66139 

0.7 

0.0344 

0.05U7 

0.0927 

0.094 

0.1098 

0.1770 

0.0288 

0.0298 

0.48 

0.30961 

0-1035 

0-1498 

0-0912 


to 0 033 
to 0-0293 
to 0-0514 
to 0-188 


to 0-0335 


to 0-0949 
to 0-1105 


to 0-6 


Volume, 
air = 1 . 
1-000 
0-9033 
0-9764 
1-0000 
1-034 
1-2583 
1-3505 
1-553 
1-96 


Weight, 
air = 1. 


1-000 

12-34 

0-8848 

1-0318 

1-0805 

0-828 

0-8878 

1-5763 

3-136 


0035 


0-0355 

0-0547 

0-1015 

0-1013 

0-1218 

0-19 


Weight, 
water = 1 . 

0-2669 

3-2936 

0-2361 

0*2754 

0-2884 

0-221 

0-2369 

0-4207 

0-847 


Capucit;I for Caloric is the relative ability of bodies to retain the specific caloric. 
Capacity for caloric is Inverse as the density of the substances. The specific 
caloric multiplied by the atom weight of a substance, gives the constant number 
0-375 (average) which proves that the atoms have equal capacity for caloric in 
.all substances. This is a fact with no known reason, but by it valuable results 
may be opened. 

Table of Relative Capacity for Caloric* 


Names. 

Equal 

Equal 

Names. 

Equal 

Equal 

Weights. 

Volume. 

Weights. 

Volume. 

Water 

1-000 

1-000 

Zinc 

0102 


Copper 

0-114 

1-027 

Tin 

0-060 


Iron 

0-126 

0-993 

Lead 

0-043 

0-487 

Brass 

Gold 

0-116 

0-050 

0-971 

0-966 

Glass 

0-187 

0-443 

Silver 

0-0S2 

0-833 



. 


When the volume diminishes the capacity for caloric will also be diminished 
and thus part of the caloric will profuse the body. A volume of air 'ompressed 
to 1 its bulk will fire tinder, which requires a temperature of about 550°. 


2'J 






























230 


Steam. 


STEAM. 


Steam is the vapour into which water is converted by the appli¬ 
cation of heat. 

Let AB be a cylindrical glass tube in which is fitted a piston a of 
one Square inch area; consider this piston to have no friction or 
weight, and can be moved steam-tight from A to B. Let the 
tube be 1728 inches from A to B, the space under the piston a just 
one inch from the bottom being filled with water of 32° Fah., which 
will be one cubic inch; weigh the whole apparatus. Now, place a 
lamp under the tube in a position as represented by the Figurfe, 
and notice the time, (say 10A f 5w.) The temperature of the watel? 
will gradually increase, and the piston a maintain a contact with ft 
until the water begins to boil, which time is to be carefully noticed; 
now (10/i., tbm.) It will be found that temperature of the water has 
raised from 32° to 212°, which took 10/i, 15 m — 10/t bm = 10 1 
minutes. 

Let the lamp still remain and the boiling be continued. The 
piston a will now leave the water, and gradually ascend towards B, 
apparently leaving a space between itself and the water, the latter 
will gradually diminish as the piston ascends, which indicates that 
steam is gradually formed, and occupies the space between the water 
and the piston, and' as the piston has no weight or friction it is 
evident that the density of the steam must be the same as the sur¬ 
rounding atmosphere. 

But another important faculty of steam and water will now be- 
manifest, namely, that the temperature of both will remain the" 
same, 212°, as at the boiling point, (10/i 15 m,) consequently the heat 
from the lamp which goes into the water and steam is not sensible 
but becomes latent. The water is now getting very low; observe C 
carefully the moment when it apparently disappears on the bottom A 
of the tube. - - “ now ” (11A 10 m..) The piston a will be found at 
B, 1700 inches from A, and the time from the boiling-point is 
(11/i 10m)— (10/i 15m) = 55 minutes = 5j times that occupied to 
raise the water from 32° to 212° •= 180°; hence the quantity of heat 
from the lamp now contained in the steam is 180X54+180 = 1170° 
of which 180° is sensible and 990 latent. If the water had been en¬ 
closed in a vessel to prevent evaporation, and the same quantity of 
heat 1170° imparted to it, it would have a temperature of 1202, 
which is about that of metals when red hot in daylight. 

Again to the tube A B, at the time before noticed, viz., 11/i, 10m. 
j Take the lamp away, weigh the apparatus, and it will be found the 
I same weight as before ; hence, the same quantity of water is still in 
l the t ube, but in the form of steam. The heat will now radiate from ^ 
the tube, and it will be observed that the piston a gradually 
descends towards A, and the inner surface of the tube will be cov¬ 
ered with a dew which will soon fall to the bottom as water, but still 
maintain the heat of 212°, until the piston a has fully reached its 
former position at A, when the same quantity of water (one cubic 
inch) will occupy the same space as before the lamp was put under 
it, but with a temperature of 212°. 

The heat required to make steam of one cubic inch of water is able 
to raise 5£ +l cubic inches from 32° to 212°; or steam at 212°, formed 
of one cubic inch of water, can raise 5£ cubic inches of ice cool water 
from 32° to 212°, when mixed together, making 6£ cubic inches. 



<a> 

t- 

r»- 





Effect of Steam. 

By the preceding experiment we find that one cubic inch of water 
will be 1700 cubic inches converted into steam; or one cubic inch of 
water makes one cubic foot of steam 212°, the same density as the 
surrounding atmosphere which is 14J pounds per square inch ; the 
effect of steam in the experiment was consequently a weight of 
F = 14j pounds raised 1700 inches = 142 feet in 55 minutes, or 


























Steam. 


231 


0-635 Effects. 
t 

See Eormula 5, page 148. 

Advantage of Using Higli Steam* 

Let us now make the same experiment with the tube AB, and load the piston 
with 14# pounds, which will be a weight F = 29'5 pounds including the atmos¬ 
phere. Set the lamp under as before, and the experiment is in operation. The 
temperature of the water will now not cease to increase when it has attained 
212°; nor will the piston a begin to raise after 10 minutes as in the former 
experiment; but, when the w ter has attained 250°, it will cease to increase, the 
piston commence'to ascend, and steam to generate. The piston will now only 
raise 930 inches from A, which will occupy the same time as before. The me¬ 
chanical effect of the steam is therefore 29'5 pounds raised 930 = 77'5 feet in 55 
minutes, or, 

29*5X77-5 


P = 


55X60 


== 0-693 Effects, 


which exceeds the former experiment about 9 per cent. 


= 0 913 01100 


91*3 =*= 8-6 per cent. 


an advantage of using higher steam. 

This per centage will increase as the steam is used higher. 

Advantage of Using Steam Expansively# 

We now continue the latter experiment. The piston a stands at 930 inches 
from A ; take the lamp away, and remove the 14# pounds on the piston. The 
steam of 25U° in the tube will now raise the piston a to B, at 1700 inches from 
A and the temperature of the steam will decrease from 250° to 212°; conse¬ 
quently the same steam has produced an additional effect by raising 14J pounds 
(the pressure of the atmosphere.) 1700 — 930 = 770 inches high = 64-165 feet, 
which for comparison, will here assume to be accomplished in the same time 
55 minutes, we shall then have 


14-75X64-165 

55X60 


= 0-29127 Effects 


and 070469+0-29127 = 0-99596 Effects produced by the same quantity of steam, 
0-64363 


and 


~ = 0-646. 


9-99596 

gained by using the steam high and expansively, 


100 — 64*6 = 35"3 per cent., 


Advantage taken of the Pressure of the Atmosphere hy 

Vacuum, 

Again to the latter experiment, the piston a stands at B , and the tube is full 
of steam at 212°; let there now be introduced among the steam 5| cubic inches 
of ice cool water, (32°), the steam will immediately condense to water, and the 
piston a begin to descend; finally, between the piston and the bottom of th> 
i tube will be found 6£ cubic inches of water at 212°, hence the atmospheric 
! pressure has reproduced an effect equal to that the steam before expended on 
it, or 0-64363 effects. 

| The principal features of the application of steam to produce mechanical 
effects are now illustrated, and we will proceed to give the principal Rules, For¬ 
mulas, and Tables, respecting its property. 

If steam is reduced in volume, its density and temperature will increase; and 
when additional heat is applied to steam its density or volume will increase the 
same as if it was produced direct from water. 

















232 


Taele of Properties of Steam. 


p-r 

Atmosph .. 

included. 


Tempera- 


Pound* 

Specific 

ture 

of Steam' 

Mercuiy. 

per square 
iuch. 

gravity, 

air = 1. 

32° 

0-200 

0-098 

0-0041 

40 

0-263 

0-129 

0-0053 

50 

0-375 

0-184 

0-0074 

60 

0-524 

0-257 

0-0102 

1 70 

0-721 

0-353 

0-0136 

80 

1-000 

0-490 

0-0186 

90 

1-36 

0-666 

0-0250 

100 

1-86 

0-911 

0-0333 

103 

2-04 

1-000 

0-0364 

i 110 

2-53 

1-240 

0-045S 

120 

3-33 

1-632 

0-0576 

130 

4-34 

2-129 

0-0538 

140 

5-74 

2-813 

0-0960 

145 

6-53 

3-100 

0-10S 

150 

7-42 

3-636 

0-122 

155 

8-40 

4-166 

0-137 

160 

9-46 

4-635 

0-153 

165 

10-68 

5-23 

0-171 

170 

12-13 

5-94 

0-193 

175 

13-62 

6-67 

0-215 

180 

15-15 

7-42 

0-238 

185 

17-00 

8-33 

0-265 

190 

19-00 

9-310 

0-294 

195 

21-22 

10-40 

0-325 

200 

23-64 

11-58 

0-36 

205 

26-13 

12-80 

0-394 

210 

28-84 

14-13 

0-431 

211 

29-41 

14-41 

0-440 

212 

30-00 

14-70 

0-448 

212-8 

30-60 

15- 

0-457 

214-5 

31-62 

15-5 

0-471 

216-3 

32-64 

16- 

0-484 

218- 

33-66 

16-5 

0-497 

219-6 

34-68 

17- 

0-512 

221-2 

35-70 

17*5 

0-529 

222-7 

36-72 

18- 

0-540 

i 221-2 

37-74 

18-5 

0-554 

i 225-6 

38-76 

19- 

0-567 

227-1 

39-78 

19-5 

0-581 

228.5 

40-80 

20- 

0-595 

229-9 

41-82 

20-5 

0-608 

! 231-2 

42-84 

21- 

0-612 

I 232-5 

43-86 

21-5 

0-636 

! 233-8 

44-88 

22- 

0-65 

j 235-1 

45-90 

22-5 

0-663 

236-3 

46-92 

23- 

0-677 

237-5 

47-94 

23-5 

0-690 

238-7 

48-96 

24- 

0-704 

239-9 

49*98 

24-5 

0-717 

241* 

51-00 

25- 

0-730 

243-3 

i 

53-04 

26- 

| 0-756 

I 


k 


Atmosphere excluded- 

Volume 

compared 

Number 
of atmos- 

Inches of 

Pounds per 

with water. 

pheres. 

Mercury, 

square inch. 

187407 


—29.79 

—14-60 

144529 

0-01 

-29-73 

-14-57 

103350 

0-01 

-29-62 

-14-52 

75421 

0-02 

-29-47 

14*44 

55862 

0-02 

-29-27 

-14-35 

41031 

0-03 

-29-00 

-14.21 ! 

30425 

0-05 

-28-63 

-14-03 

22873 

0-06 

-28-13 

-13.79 

20958 

0-07 

-27-95 

-13.70 

16667 

0-08 

-27-46 

-13 46 

13215 

0-11 

-26-66 

—13*07 

10328 

0-14 

-25-65 

-12*57 

7938 

0-19 

-24-25 

-11*89 

7040 

0-22 

-23-46 

-11*60 

6243 

0-25 

-22-57 

-11*06 

5559 

0-28 

-21-59 

-10 54 

4976 

0-31 

-20-53 

—10'07 

4443 

0-35 

-19-31 

- 9 47 

3943 

0-4 

-17-86 

- 8*76 

3538 

0-45 

-16-37 

- 8*03 

3208 

0-50 

-14-84 

- 7‘28 

2879 

0-56 

-12-99 

- 6‘37 

2595 

0-63 

-10-99 

- 5'39 

2342 

0-71 

- 8-77 

- 4’30 

2118 

0-79 

- 6-35 

- 3*12 

1932 

0-87 

- 3-86 

- 1*90 

1763 

0-96 

- 1-15 

- 0*57 

1730 

0-98 

4 0-58 

- 0'29 

1700 

1-00 

~ o-oo 

4 O'OO 

1669 

1-02 

4 0-60 

4 0-30 

1618 

1-05 

4 1-62 

4 0 80 

1573 

1-09 

4 2-64 

4 1 30 

1530 

1-12 

4 3-66 

4 1*80 

1488 

1-15 

4 4-68 

4 2'30 

1440 

l *19 

4- 5-70 

4 2-8 

1411 

1-22 

4 6-72 

4 3-3 

1377 

1-25 

4 7*74 

4 3-8 

1343 

1-29 

4 8-76 

4 4*3 

1312 

1-33 

4 9-78 

4 4-8 

1281 

1-36 

410-80 

4 5-3 

1253 

1-40 

411-82 

4 5-8 

1225 

1-43 

412-84 

4 6-3 

1199 

1-46 

413-86 

4 6-8 

1174 

1-50 

414-88 

4 7-3 

1150 

1-53 

415-90 

4 7-8 

1127 

1-56 

416-92 

4 8-3 

1105 

1-60 

417-94 

4 8-8 

1084 

1-63 

418-96 

4 9-3 

1064 

1-67 

419-98 

4 9-8 

1044 

1-70 

421-00 

410-3 

1007 

| 1-77 
: 

423-04 

411-3 































TABLE OF PROPERTIES OF ‘STEAM. 233 



Aimosph . included . 


k 


Atmosphere excluded. 

Tempera¬ 

ture 

laches of 
Mercui . v . 

Pound* 
per square 

Specific 

gravity. 

Volume 

compared 

Number 
of atmos 

Inches of 

Pounds per 

of Steam, 

Inch. 

air = 1« 

with water. 

pheres. 

Mercury. 

square iuci:. 

245 * 5 P 

55-08 

27 

0-784 

973 

1-83 

+ 25-08 

+ 12-3 

247-6 

57-12 

28 

0*810 

941 

1-90 

+ 27-12 

+ 13-3 

249-6 

59-16 

29 

0-836 

911 

1-97 

+ 29-16 

+ 14-3 

261-6 

61-20 

30 

0-863 

883 

2-04 

+ 31-20 

+ 15-3 

253-6 

63-24 

31 

0-889 

857 

2-11 

+ 33-24 

+ 16-3 

255-5 

65-28 

32 

0-915 

833 

2-18 

+ 35-28 

+ 17-3 

257-3 

67-32 

33 

0-941 

810 

2-24 

+ 37-32 

+ 18-3 

259-1 

69-36 

34 

0-968 

788 

2-31 

+ 39-36 

+ 19-3 

260-9 

71-40 

35 

0-993 

767 

2-38 

+ 41-40 

+ 20-3 

262-6 

73-44 

36 

1-020 

748 

2-45 

+ 43*44 

+ 21-3 

264-3 

75-48 

37 

1-045 

729 

2-52 

+ 45-48 

+ 22-3 

265-9 

77-52 

38 

1-071 

712 

2-59 

+ 47-52 

+ 23-3 

267-5 

79-56 

39 

1-097 

695 

2-65 

+ 49-56 

+ 24-3 

269-1 

81-60 

40 

1-122 

679 

2-72 

+ 51-60 

+ 25-3 

270-6 

83-64 

41 

1-148 

664 

2-79 

+ 53-64 

+ 26-3 

272-1 

85-68 

42 

1-175 

649 

2-86 

+ 55-68 

+ 27-3 

273-6 

87-72 

43 

1-200 

635 

2-92 

+ 57-72 

+ 28-3 

275 - 

89-76 

44 

1-225 

622 

3-00 

+ 59-76 

+ 29-3 

276-4 

91-80 

45 

1-249 

610 

3-06 

+ 61-80 

+ 30-3 

277-8 

93-84 

46 

1-275 

598 

3-13 

+ 63-84 

+ 31-3 

279-2 

95-88 

47 

1-567 

586 

3-20 

+ 65-88 

+ 32-3 

280-5 

97-92 

48 

1-325 

575 

3-26 

+ 67-92 

+ 33-3 

281-9 

99-96 

49 

1-351 

564 

3-32 

+ 69-96 

+ 34-3 

283-2 

102-0 

50 

1-376 

554 

3-40 

+ 72-00 

+ 35-3 

284-4 

104-0 

51 

1-400 

544 

3-47 

+ 74-00 

+ 36-3 

285’7 

106-1 

52 

1-426 

534 

3-53 

+ 76-1 

+ 37*3 

286-9 

108-1 

53 

1-450 

525 

3-60 

+ 78-1 

4 38-3 

288-1 

110-2 

54 

1-477 

516 

3-67 

+ 80-2 

39-3 

289-3 

112-2 

55 

1-500 

508 

3-74 

+ 82-2 

+ 40-3 

290-5 

114-2 

56 

1-523 

500 

3-81 

+ 84-2 

+ 41-3 

291-7 

116-3 

57 

1-548 

492 

3-88 

+ 86-3 

+ 42-3 

292-9 

118-3 

58 

1-575 

484 

3-94 

+ 88-3 

+ 43-3 

294-2 

120-4 

59 

1-598 

477 

4-01 

+ 90-4 

+ 44-3 

295-6 

122-4 

60 

1-621 

470 

4-08 

+ 92-4 

+ 45-3 

296-9 

124-4 

61 

1-646 

463 

4-15 

+ 94*4 

+ 46-3 

298-1 

126-5 

62 

1-671 

456 

4-22 

+ 96-5 

+ 47-3 

299-2 

128-5 

63 

1-698 

449 

4-28 

+ 98-5 

+ 48-3 

300-3 

130-5 

64 

1-719 

443 

4-35 

+ 100*5 

+ 49-3 

301-3 

132-6 

65 

1-743 

437 

4-42 

+ 102-6 

+ 50-3 

302-4 

134-6 

66 

1-755 

434 

4-49 

+ 104-6 

+ 51-3 

303-4 

136-7 

67 

1-794 

425 

4-55 

+ 106-7 

+ 52-3 

304-4 

138-7 

68 

1-818 

419 

4-62 

+ 108-7 

+ 53-3 

305-4 

140-8 

69 

1-839 

414 

4-69 

+ 110-8 

+ 54-3 

306-4 

142-8 

70 

1-868 

408 

4-76 

+ 112-8 

+ 55-3 

307-4 

144-8 

71 

1-891 

403 

4-82 

+ 114-8 

+ 56-3 

308-4 

146-9 

72 

1-915 

398 

4-89 

+ 116-9 

+ 57-3 


148-9 


1*938 

393 

4*96 

5*03 

+ 118-9 

+ 58-3 

out? o 

310-3 

151-0 

4 t) 

74 

1-963 

3 >8 

+ 121-0 

- r - 59-3 

311-2 

153-0 

75 

1-991 

383 

5-09 

+ 123 * 

+ 60-3 

312-2 

155-1 

76 

2-011 

379 

5-17 

+ 125-1 

+ 61-3 

313-1 

157-1 

77 

24)36 

374 

5-23 

i . ! 

4 - 127-1 

. 1 

+ 62-3 




- 1 



20 * 

























234 


Table of Properties of Steam. 



Atmosph. included 


k 

r 

Number 
of atmos- 

Atmosphere excluded. 

Tempera¬ 

ture 

Inches of 

Pounds 

Specific 

gravity, 

Volume 

compared 

Inches of 

Pounds per 

of Steam. 

Mercury. 

Inch. 

air = 1. 

with water. 

pheres. 

Mercury. 

square inch. 

314-0° 

159-1 

78 

2-060 

370 

5-30 

+ 129-1 

+ 63-3 

314-9 

161-2 

79 

2-081 

366 

5-37 

+ 131-2 

+ 64-3 

315-8 

163-2 

80 

2-105 

362 

5*44 

+ 133-2 

+ 65-3 

316-7 

165-3 

81 

2-128 

358 

5-51 

+135-3 

+ 66-3 

317-6 

167-3 

82 

2-152 

354 

5-57 

+ 137-3 

+ 67-3 

318-4 

169-3 

83 

2-178 

350 

5-64 

+139-3 

+ 68-3 

319-3 

171-4 

84 

2-203 

346 

5-71 

+ 141-4 

+ 69-3 

320-1 

173-4 

85 

2-228 

342 

5-78 

+143-4 

+ 70-3 

321-0 

175-5 

86 

2-248 

339 

5-85 

+145-5 

+ 71-3 

321-8 

177-5 

87 

2-275 

335 

5-91 

+147-5 

+ 72-3 

322-6 

179-6 

88 

2-295 

332 

5-98 

+149-6 

+ 73-3 

323-5 

181-6 

89 

2-322 

32'8 

6-05 

+ 151-6 

+ 74-3 

324-3 

183-6 

90 

2-343 

325 

6-12 

+153-6 

+ 75-3 

325-1 

185-8 

91 

2-365 

322 

6-19 

+155-6 

+ 76-3 

325-9 

187-8 

92 

2-389 

319 

6-26 

+157-8 

+ 77-3 

326-7 

189-8 

93 

2-411 

316 

6-32 

+159-8 

+ 78-3 

327-5 

191-9 

94 

2-435 

313 

6-39 

+161-9 

+ 79*3 

328-2 

193-9 

95 

2*459 

310 

6-46 

+163-9 

+ 80-3 

329-0 

196-0 

96 

2-483 

307 

6-53 

+166-0 

+ 81-3 

329-8 

198-0 

97 

2-505 

304 

6-60 

+168-0 

+ 82-3 

330-5 

200-0 

98 

2-530 

301 

6-66 

+170-0 

+ 83-3 

331-3 

202-0 

99 

2-558 

298 

6-73 

+172-0 

+ 84 "3 

332-0 

204-0 

100 

2-583 

295 

6-80 

+ 174-0 

+ 85-3 

335-8 

214-2 

105 

2-703 

282 

7-13 

+ 194-2 

+ 90-3 

339-2 

224-4 

110 

2-815 

271 

7-47 

+ 194-4 

+ 95-3 

342-7 

234-6 

115 

2-947 

259 

7-82 

+204-6 

r 100-3 

345-8 

244-8 

120 

3-036 

251 

8-15 

+214-8 

+105-3 

349-1 

255-0 

125 

3-178 

240 

8-5 

+225-0 

+ 110-3 

352-1 

265-2 

130 

3-270 

233 

8-83 

+235-2 

+ 115-3 

355-0 

275-4 

135 

3-405 

224 

9-16 

+245-4 

+ 120-3 

357-9 

285-6 

140 

3-497 

218 

9-51 

+255-6 

+ 125-3 

360-6 

295-8 

145 

S-626 

210 

9-83 

+265-8 

+ 130-3 

363-4 

306-0 

150 

3-712 

205 

10-2 

+ 276-0 

+ 135-3 

368-7 

326-4 

160 

3-941 

193 

10-9 

+296-4 

+ 145-3 

373-6 

346-8 

170 

4-028 

183 

11-5 

+316-8 

+ 155-3 

378-4 

867-2 

180 

4-375 

174 

12-2 

+337-2 

+ 165-3 

382-9 

387-6 

190 

4-585 

166 

12-9 

+357-6 

1+175-3 

387-3 

408-0 

200 

4-82 

158 

13-6 

+378-0 

1+185-3 

403-8 

509- 

250 

5-90 

129 

17-0 

+479- 

+235-3 

420-3 

612- 

300 

7-00 

109 

20-4 

+582- 

+285-3 

435-0 

714- 

350 

8-00 

953 

23-8 

+684- 

+345-3 

446-5 

816- 

400 

8-95 

851 

27-2 

+ 786- 

+385-3 

471-3 

1019- 

500 

10-9 

700 

34-0 

+989- 

+485-3 

487-0 

1223- 

600 

12-8 

597 

40-8 

+1193- 

+ 585-3 

519- 

1631- 

800 

15-7 

486 

54-4 

+ 1601- 

+785-3 

548. 

2038- 

1000 

19-7 

387 

68 e 

I 

+2008- 

+985-3 

f 

J 























Steam. 


•235 


Letters denote. 

F— force of the steam, or pressure per square inch in pounds. 

I — inches of Mercury that balances the steam. 

T = temperature of the steam in degrees of Fahrenheit’s Thermometer. 


Formulas for Steam above 212°. 

■ ■ 

I= (l55 +0 ' 584 )’ ' ’ 

T=( 0-52)202, 


1 , 


3. 


Example 1. The temperature of a quantity of steam is found to be 275°. 
Required the density in pounds per square inch ’ 

/275 , 

F— 1 202 +0*52 1 = 44-3 pounds per square inch. 


By logarithms, 


275 

^—+0-52 = 1-881. 
202 - 

log.1-881 = 0-274389 
6 


log.44-3 = 1-646334 or 44-3 pound per square inch. 

The properties of steam are calculated and contained in the accompanying 
Table, as noted on the top. The two last columns contain the inches of mercury 
and pressure per square inch commonly expressed in practice; it is O at the 
temperature 212°, and below that temperature it is negative, which denotes so 
much vacuum. If the temperature in a condenser is 120, the vacuum is 13-07 
pounds. 

To Find the Weight of Steam, 

RULE. Multiply the specific gravity of the steam by the weight of one cubic 
foot of air = 0-07529, and the product is the weight per cubic foot of the steam 
in pounds. 

To Find the Quantity of Water of which a given quantity of Steam has been, or can 

be produced. 

RULE. Divide the cubic contents of the steam by the volume h in the Table, 
and the quotient is the cubic content of the Water. 

Force or Feed Pumps, 

Letters denote. 

d ~ diameter \ of tJie f orce pump, single acting. 

5 —— StiOKcj j 

*S ~ 2STI s ^ eam blinder piston, in inches, double acting. 

Jc = volume given in the Table at the given pressure of steam. 

The stroke of the steam piston is only that under which steam is admitted to 
the cylinder. 


d 


= 20 \/r,- 


5 = 4 


D*S 

led** 


4, 5, 


Slip water included, 








206 


ATR-PtfMP. 


Example. Required the diameter of a force-pump having the same stroke as 
the cylinder piston s = 38 inches, diameter of cylinder D = 30 inches, the steam 
is cut off at ± the stroke, and the steam pressure + 50 pounds per square inch v 
Here k = 437, and £ — 19 inches, because steam is cut off at £ the stroke. 


d = 2X30 


\/ 


19 

437X38 


= 2-03 inches. 


To find the Quantity of Condensing Water. 

Letters denote. 

q = condensing water of temperature t, in cubic feet. 

Q == steam of temperature T, in cubic feet. 
k — volume in the Table. 

t' — temperature in the condenser when the steam and water are mixed. 

"• * : r-4 (2(990 + 7- V ) 

q k(t' — t) * 

Dimensions of the Air Pump* 

s = stroke 61 " | of tlie air P um P> sin S le Acting. 

I) — diameter' 


6 , 


S — stroke } s ^ eam cylinder, double acting. 


! - s V 

= 50° 

+ 


,87990 + 7’— t') 
k s(t’ — t) ’ 

Assume if = 100°, and t = 50°, we shall have, 
d = 0-326Z+ 


£(9 40 + T) 
k s ’ 


led* ’ 


single acting air-pumps. 


d== 0*i 


■rns/m 


£(940 + T) 

s 


s = 0-053D* 


£(940 + T) 
k d* ’ 


double acting air pumps. 


7, 


8, 


9, 


10 , 


11 . 


Example. A single acting air-pump is to be constructed for an engine 
D = 38 inches, £ = 45 inches stroke of the cylinder; the stroke of the air- 
pump can be 32 inches, and the exhaust steam is 261°. Required the diameter 
of the air-pump ? fc = 767. 


d = 0-32&X38 


V 


*Bsr—— 


^©“Slip water included. 7’and k must be taken for the exhaust steam, as the 
steam may have had worked expansively; the area of the foot valve must be 
calculated from the following formulas. 

Foot Valve in the Air Pump. 

To render an air-pump to work well, and with the greatest advantage, it is 
necessary to pay particular attention to the following formulas. The force by 
which the water is driven from the condenser through the footvalve into the 
air-pump is limited by the pressure in the condenser; this pressure is the 
vacuum subtracted from 14-7 pounds; it is noted in the third column where 
the temperature in the condenser is opposite, in the first column. Every 
pound of this pressure per square inch balanced a column of water 27 inches 
liigh, which is the head that presses the water from the cond nser. 





















Air-Pump. 


237 


Letters denote. 

m — area of the air-pump piston, 
a = area of the foot-valve, or bucket-valve. 

= diameter of the air-pump-piston, 
tl = diameter of the foot-valve, when round. 


= stroke of air-pump piston, in feet. 

33 = pressure in the condenser at the temperature T. 
n = number of strokes of the air-pump piston per minute. 


& 3s n 

100 VW 

12, 

^3\/ 3! « 

* ~ lo VW ’ 

15, 

^ lOOWi 

n& 9 

13, 

a _iooW$ 

n ’ 

16, 

n _l0(W5 
& S 

14, 

n __ 100 W 31 

W 3 ’ 

17. 


Example. The area of an air-pump-piston is 2‘35 square feet, stroke of 
oiston 5^ = 3 - 6 feet, to make n = 40 strokes per minute, and the pressure to be 
^3 = 3-2 pounds. Required the area of the foot-valve. 


a = 


2-35X3'6X40 

100 v'TxT 


= T85 square feet. 


To Find f.lie Velocity and Quantity of tlie Injection Water 
through the Adjustage into the Condenser* 

Letters denote. 


v = velocity in feet per second. 

h = head of the press water; + when above, and— below the adjustage. 

F= vacuum, noted — or negative in the last column, but is positive in the 
formulas. 

q = quantity of water discharged in cubic feet, per second. 
a — area of all the holes in the adjustage in square feet. 

^ ~ le*nyth^ er } t * ie injection pipe, in feet. 

n = double strokes of cylinder-piston, or revolutions per minute. 

A, D, and S, dimensions of the steam cylinder, in feet. 

T = temperature, and k = volume coefficient of the exhaust steam. 


5^ r 2F±h’ 


t = 8 s/2 F±h 


IB, 


19, 


n S 2^(940+7 1 ) 2Q 
55 k 


q — 5a F±h, . 

d - °- 35 s/M’ 

n S D a (940+ T) 
a ~ ZI5k</2F+h * 


21 , 

22 , 

23, 



























238 


Steam. 


Example. Required the diameter of an injection pipe L = 10 feet lon< 
which shall supply q = 1*3 cubic feet of water per second into a vacuum of 12 
pounds per square inch, the head of press water h = 2 feet ? 


d = 


0-35 


ln 2>-L 3 _ = 0*3055 feet = 3|I inches. 

2X12+2 


1 6 


Area of Steam Passages* 

a == area of the steam pipe, sq. in. 

A == area of the cylinder piston, sq. in. 
d = diameter of the pipe, in inches. 

D = diameter, S = stroke of cylinder, in inches. 


A Sn 
35000’ 


a== WSn 

186 


24, 25. 


Example. Required the diameter of a steam-pipe for a cylinder D = 40 
inches. Stroke of piston S = 48 inches, and n = 38 revolutions per minute ? 


d _ 40+48X38 
186 


& - 2 inches, nearly. 


Steam Ports to the Cylinder# 

A Sn 


a 


30000’ 


26, 


Safety Valve* 

Three-fourths of the fire grate in square feet is a good proportion for the 
safety valve in square inches. 

Notation of Letters corresponds with Figure 3, Plate VIII. 
a = area of safety valve in square inches. 

P = pressure per square inch in the boiler 4 

TF= weight on the safety valve lever >in pounds. 

Q = weight of the safety valve and lever j 
l = lever for W 1 
e = “ aP >-in inches. 

*= « Q) 

Balance the lever over a sharp edge, and the centre of gravity Q is found; 
measure the distance x from the fulcrum C. 


a P 

e = W l+Q v 

27, 

cl JP € — Q X 
i 

29, 

P - 

WlfQx 

28, 

, aPe — Q x 

CO 

p 


a e 


W 



Example. Area of the safety valve a = 9 square inches, e = 4£ inches, 
W— 50 pounds, weight of the lever and safety valve Q = 15 pounds, and a; = 17 
inches. Required at what distances l, V and L" will the weight W indicate pres¬ 
sures of P — 30, P' — 40, and P" — 50 pounds ? 

= 9X30X4*5+1 5X17 = 2g . 2 in 

50 ’ 

from the fulcrum C the weight TFwill indicate P= 30 pounds, 

V = 37-9 inches, when P' — 40 pounds. 

I" = 45-8 “ “ P" = 50 “ 

and thus the lever can be graduated. 
























Steam. 


239 


11 


Cut-off Steam, 

In order to pave steam, or more correctly to employ it's effect to a higher 
degree, the admittance of steam, to the cylinder is shut off when the piston has 
moved a part of the stroke; from the cut-off point the steam acts expansively 
with a decreased pressure on the piston, as represented by the accompanying 
Figure. g 06> 

Let the steam be cut off at } of the stroke, 
and .da represent the total pressure, say 20 
pounds per square inch which will continue to 
the point E where the admittance of steam is shut 
off at | the stroke S. The steam Act cE. is now 
acting expansively on the piston, and the 
pressure decreases as the volume increases, when 
the piston has attained Cc or f of S, the pressure 
C'c — 10 pounds, only half the pressure Aa = 20 
because the volume A a e E is only half of A a c C, 
and bo on until the piston has attained Bb the 
pressure B'b = |X20 = 6 - 66 pounds. 

The mean pressure, or the effectual pressure, 
throughout the stroke, will be about 13'33 pounds 
per square inch, or 66 per cent., but the quantity 
of steam used is only 33 per cent., hence 33 per 
cent, is gained by using the steam expansively. 



Letters denote. 

J =c the part of the stroke S with full steam. 

P = pressure per square inch under full admittance of steam. 
F = the effectual force per square inch throughout the stroke 
p = pressure per square inch at the end of the stroke 


p — —, and F - 
S 


^ 2-3(log.S-log.l)+l^~. 


Example. Stroke of piston is S = 6'4 feet, and the steam is cut off at t — 2 - <S 
feet, the pressure under fall steam is P = 50 pounds, Required the effectual 
pressure F — ? 

From log.6 - 4 = 0-806180 

Subtract log.2-8 = 0447158 


P= 


2 - 3 X 0359622 + 1 = 1’69 nearly. 
50X1-69X2-8 


6'4 


37 pounds. 


Table of Expanded Steam. 


F. 

p. 

0-3845 P 

0-125P 

0-597 P 

0-25P 

0-698 P 

0-33P 

0-741 P 

0-375P 

0-844 P 

0-6 P 

0-918.P 

0-625P 

0-935 P 

0-666P 

0-965P 

0-75P 

0-991P 

0-875P 


& - 

«© 3 

^ # 

c/a JL 

a # 


Example. The pressnre in the hoiler is P = 68 pounds ; cut off the steam at 
| of the stroke. Required the effectual pressure throughout the stroke, and 
density of the exhaust steam? 

! F= 0‘741X68 = 50-4 pounds effectual pressure. 

p = 0-375X68 = 25-6 pounds density of exhaust. 




















240 


Rude Valves. 


SLIDE VALVES. 

The slide valve motion is one of the most important features in causing a 
steam engine to work well, and to employ the effect of steam economically. 
The author of this hook being well acquainted with disarrangements on this 
point, has here endeavoured to give a good working-drawing of the proper pro¬ 
portions and arrangements of slide-valve motion. (See Plate VIII •) 

Main Valve* 

It will be best to assume a certain size cylinder, and at the same time give the 
proportions for any size. 

D = 34 inches, diameter of the cylinder. 

S = 18 inches stroke of piston* 
n = 56 double strokes per minute. 

We have the area of the steamports to, from Formula 26, page 238. 


3 4»X0-785X18X56 

30600 


= 30 square inches, nearly. 


to = 


D+S 34+18 


26 


26 


= 2 inches, 


the width of the steamport; if the quotient gives a fraction take the nearest 
quarter or eighth. 

a 30 

— = — = 15 inches, breadth of steamport. 

to 2 

r = £ to about = 1 inch, the exhaust port o = 2to — Jr = 3J inches, and 
/ = o + 2r — 5£ inches, h — f — $r = 5y inches, k = 1 Jto = 3 inches, and 
i - h-\-2k — 11^- inches, e = to = 2 inches. 

* The stroke and diameter is here rather out of proportion, but we will maintain 
them in the calculations as they suit the drawing, which is purposely made to 
show the slide valves on a large scale. The rules will however suit any propor¬ 
tions of diameter and stroke. 

To Find llie Stroke of the Eccentric* 

s = stroke of the eccentric in inches. 
s — i — f — x r = 5J inches. 

The lap L — \(i —/— 2to) = } inches. 

The lead of the valve, or opening of the steamport when the crank pin stands 
the centre should be about 


l 


mx'n 2]/56 ... , 

-§jj- = = i inches, nearly. 


Having finished the main valve and ascertained the stroke of the eccentric, 
it is now required to find the position of the centre b, (Plate VI.,) of the eccentric, 
to the crank-pin. Suppose the crank pin of the engine stands at a on the centre 
nearest to the cylinder, and the eccentric rods are attached direct to the valve 
rods; draw the line dd , at right-angle to the centre-line aa" of the engine, 
then 

the angle, sin.IT= 2 ^±5 = = 0-409, or W = 24° 10'. 

£ Of 

See Plates VIII and IX. 

To Find the position of the Crank-Pill at the moment the 

Main Valve opens* 


y 


s cos 


— 78X6 25 — inches nearlv 
L W~ 5-5X0 9123 U eS ’ “early, 


} from the centre line. 

i 

l _ 













J 


' 

:• -• - ' 

; - ■ ■■•'. ;i; 


. Y ' r - 


! 1 'Y "■ ' ,r i 



















£-■ • 








U ■ 




' —. ■ 







- ■ r i ■ • I 




~ - 


. • 


' 



















p/t/Us m 





J.W.A'ys train 


















































































































































































































































































Eccentrics. 


Pla te LK. 










































-- ‘ ■ - 








>'■ f •:* & h: >f ■■ -< i. • R‘ r - : 










Slide Valves. 


241 


To Find the position of the Crank at the moment the 

Exhaust opens. 

* = j (sin. IT— j(/— h)^ = j (o-409 -- ^(5*5 — 5-25)^ = 3*27 inches 
from the centre line. 

To Find the position of the Crank Pin when the Main 
Valve cuts oif the Steam* 



2S L __ 2X18X1 
s 6-5 


= 5-727 inches. 


To Find at what part of the Stroke the Main Valve Cuts 

oif the Steamy 

4 L* /2V-\ a 

Will cut off at — 1-- 1 — I -— 0 1 = 0-899 of the stroke. 

s a \ 6*5 / 

The greater the lap is, the sooner will the main-valve cut off, but if the lap is 
increased the stroke of the eccentric must also be equally increased. It does 
not work well to cut off much by the main-valve, especially when the engine 
works fast; for very slow motion it may answer to cut off at £ the stroke. 

It will be noticed that the centre of the eccentric is always ahead of the crank 
pin with an angle 90°-fw. Hence when the engine is to be reversed, the centre 
b must have the same position on the opposite side of the centre-line, or the 
eccentric must be moved forwards an angle of 90° — 2w. 


Cut-off Valve* 

The width of the cut off ports should be about d — = lj inch, and their 


breadth 


a _ 30 

2 d ~ 2XH 


= 12 inches, when two ports are used. 


Proportions of the Valve. 

a — b = c — d , a+d = 6+c, and a = 2d, and the stroke of the cut-off valve 
eccentric s = 2 b, we shall have a = 2£, b = 2£•, c = 1£, c = 1^, and 
s = 4£ inches. 

Let us assume the steam to be cut off at } = l of the stroke S, the position of 
the crank-pin a' will then be sin.w = 21 = 0-666, or u = 70° 30'; at the same 
time .he position of the centre d of the cut off l ccentric will be 


. d+c 

sin .2 = -= 

£ 


li+H 

4* 


=» 0-612, or 2 = 37° 50', 


and V — u —z = 70° 30' — 37° 50' = 32° 40', the position of the centre c when 
the crank-pin a is on the centre. This Table will show the positions of the 
centre a and c, at different cut offs. Letters correspond with Figure 1, Plate VI. 


Cut off 

at l. 

— 

V 

sin.v 

stroke of 

eccen. s. 

2 

u 

F. 

P • 


22° 10' 

0-377 

2b 

37° 50' 

60° 

0-5880 

0-250 

1 

32° 40' 

0-539 

2b 

37° 50' 

70° 30' 

0-6914 

0-333 


31° 55' 

0-527 

c-\-a 

43° 35' 

75° 30' 

0-7332 

0-375 

j i 

42° 35' 

0-675 

b-\-c 

47° 25' 

90° 

0 8350 

0-500 


46° 30' 

0-7193 

a-\-b — c 

58° 

104° 30' 

0-910 

0-625 

i 3 

5 i° 30' 

0-7933 

a-j-b — c 

58° 30' 

109° 30' 
-- 

0-985 

0-666 


It will now be observ d that the effectual pressure F in this Table is less 
than in the Table on page 239, owing to the valve not cutting off the steam 
instantly, but gradually, so that the density of the steam in the cylinder is 
already diminished at the cut off point. The valve will cut off quicker the less 
the angle z is. 

gee Figure 2, Plate VIII. The actual pressure will not form a sharp corner at 
e, or^follow the line e,e,e, as would be due when cut off at £ the stroke, but the 
line //'//will be the true diagram. Including the steam in the ports and 
steam chest, the density at the end of the stroke will correspond nearly with the 
Table. 

--- — - - - ■ - ■ - 


i 





























24:2 


Blowing-off. Salt Water. Saturation. 


BLOWING-OFF. SALTWATER. SATURATION. 

Sea water contains about 0‘03 its weight of salt. When salt water boils, fresh 
water evaporates, and the salt remains in the boiler, consequently the propor¬ 
tion of salt increases as the water evaporates, until it has reached 0-36 weight 
to the water; the salt will then commence to saturate in the boiler, and the 
water iu solution will hold 0-36 weigh t of salt to 1 of water. 

To prevent this deposit in the boiler, it is necessary to keep the salt below 
this proportion, which is overcome by withdrawing (blow off) part of the super- 
salted water, while less salted (feed water,) w. ter is replaced. It is found in 
practice that when the proportions are kept 0T2 of salt to 1 weight of water, 
the deposit will be very slight. To obtain this it will be necessary to blow off 

= 0"25 parts of the feed water, or 
0T2 ’ 

if a brine-pump is used, it should be at least 025 of the feed pump. 

IT = cubic feet of super-salted water to be blown off per minute. 

D, S, n, and k, as before, we shall have, 

Sn 

W - 

30007c * 

Example. D = 30 inches, stroke of piston 36 inches, cut off at half stroke 
S — 18, making 14 revolutions per minute, with a pressure of 30 pounds per 
square inch, k = 610. How much water must be blown off per minute f 


W = 


302X18X14 

3000X610 


= 0T24 cubic feet. 


Heat Wasted by Blowing Off. 

Letters denote. 

W= wSe! blownfff ed } in cuWc feet P er unit of time ‘ 

t = temperature of the feed water. 

T = “ “ blowing off. 

H = heat wasted, per cent. 


U = 


W(T—t) 
w(990 + T— t)' 


Example. Let the quantity of water blown off be j of the feed water, we have 
W — 1, and w — 2, the boiling point of the water will then be T = 215-5°, let 
the feed water taken from the hot-well be t — 100°. Required the quantity of 
heat lost ? 


H = 


1(215-5° — 100) 


2(990+215-5 — 100) 


= 0-052 or 5-2 ner cent. 


This is a very triflng quantity of heat lost. 


Proportion 

Boiling point 

Water blown off. 

Heat lost. 

Specific 

of Salt. 

T. 

per cent. W. 

per cent. 

gravity. 

0-03 

213-2° 

100 

100 

1-03 

0-06 

214-4 

50 

10-35 

1-06 

0-09 

215-5 

33-3 

5-2 

1-09 

0-12 

216-7 

25 

3-5 

1-12 

0-15 

217-9 

20 

2-66 

1-15 

0T8 

219- 

16-6 

2-14 

1-18 

0-21 

220-2 

14-3 

1-80 

1-21 

0-24 

221-4 

12-5 

1-56 

1-24 

0-27 

222-5 

111 

1-38 

1-27 

0-30 

223-7 

10-0 

1-23 

1-30 • 

0-33 

224-9 

9-07 

1-12 

1-33 

0-36 

226 

Water saturates. 

1-36 



















Blowing off. Salt Water. Saturation. 


213 


Heat wasted by Incrustation* 

The conducting power of iron for heat, is about 30 times that of saturated 
scales, hence a considerable portion of heat is lost when the scales become thick 


in a boiler. 


t = thickness of the scale in 16th of an inch. 
H = heat wasted, in per cent. 


II = 


f « 


32+1* 

Example. The scale in a boiler is 5 sixteenths of an inch thick. How much 
heat is lost by it ? 


H = 


5* 


0438 or 44 per cent, nearly, 


32-1-5* 

which goes out through the chimney. 

This is merely to show that the heat lost by blowing off is but trifling, com¬ 
pared with the heat lost by saturation of scales, which additionally injures the 
boiler by softening and fracturing the iron, and final explosions. 

When boilers are taken good care of by cleaning and blowing off at short 
intervals, the scales need not exceed 1 sixteenth of an inch. 


To Command tbe Engineer bow to Manoeuvre tlie Engine 

in a Steamboat; 

1 


Go ahead - 

1 

^ - 

one stroke. 

Back 

I i 

i 

two strokes. 


| 

one stroke. 

Stop - 

-0- - 

Slowly 


two short. 

Full speed 

-e-- - 

three short. 

Go ahead slowly 

1 77 
-0-0- - 

one long, two short. 

Back slowly 

i ! 77 
-#■ -0- 

two long, two short. 

Go ahead, full speed 

jin - 

one long, three short. 

Back fast - 

i I ITT 

two long, three short. 

Hurry 

rrr •? rn 

-0--e-O~ -0-0-O- 

three short repeated. 














Scbew Propellers. 


2ii 


SCREW PROPELLERS, 

Plate X., is a drawing of a Screw-propeller with proportions thus far known to 
be the most effective, particularly when the steam-engine is applied direct to 
the propeller shaft; its pitch is twice the diameter at the periphery, but con¬ 
tracts towards the centre; at the hub the pitch is lessened by the amount of 
slip assumed. 'When the propeller is geared from the engine, the pitch is gen¬ 
erally less in proportion to the diameter. 

p = i of the pitch at the periphery. 
p" = “ “ “ hub. 

s = the assumed slip in a fraction of p. 

Then p — p"+s. 

By these two pitches p end p", the helixes, acb at the periphery and dcf at 
the hub are constructed as for common screws. 

The actual pitch of the propeller at the centre of effort of the blades o is rep¬ 
resented by p' r — O’725it from the centre; or the actual pitch = 

Letters Demote, 

P = pitch of the propeller ) t ,, Derit)herv 
angle of the blades j at tbe P en P ner 7- 

D = diameter, it = radius, extreme. 

L = length parallel with the centre-line. 

m = n. uiber of blades. 

b = extreme breadth of the propeller blades over the edge, between the 
corners e, e. 

e = circle arc in the angle v. 

v = the projected angle of the blades. 

a = the projected area of the blades. 

A = the true inclined surface of the blades. 

O = acting area of the propeller. 

H= horse-power required to drive the propeller n revolutions per minute. 

Formulas for Screw Propellers, 


cot. W = _- 

7lD 


P -= cot. W jt D, - 


1 , 


p = 


360 L 

V 

7LD L 


p = 


77 D L 

V b* — L q 


360 L 

V P~’ 


3 , 

4 , 

5 , 

6 , 


b = K f v \ n Uh — L 1 7 
V 129600 ’ 7> 

' - 8, 


a = 


0'78 5D* v m 
360 ’ 


, P YY)- / t j \ 

4 - M5 (6+i) ’ ‘ 


°-5 D 1 


0 = 


V r'+n 1 £> a ’ 
D 3 if 


9 , 

10 , 


H =m<tm{ LSa * w+,yn ) 11 ’ 


_78 _ 3 /_ H 

D A/ {L Scop. TJ 


s. Tk^-0-11) 


12. 


























J.W.Xvstrom. 


































































Screw Pecpellbrs. 


245 


Example 1. The diameter of a propeller is 10 feet 6 inches, and the angle 
W = 58° at the periphery. Required the pitch P = in feet ? 

P = cot.58°X3-14XlO-5 = 20'6 feet. 

Example 2. The propeller on Plate VII. is of dimensions B — 15 feet, L = 5 
feet, W = 57° 30', the slip is 38 per cent: or £ — 0 - 38. What power is required 
to drive it 40 revolutions per minute, H == ? 


H = 


15*X40 9 


480000 


( 


5X0'38Xcos.57° 30'+0Tl 1 = 509 horses, nearly, 


) 


Example 3. A Propeller of diameter D — 12 feet, angle W == 64°, and length 
L = 3 feet 6 inches, is to he driven by a steam engine of 450 horses, the slip 
S = 0‘28. How many revolutions will it make per minute, n — t 


78 

12 


A / 1 


450 


61 revolutions 


(3*5X0'28Xcos-64°-(-0Tl) 

per minute. 

Explanation of Tables. 

Table I, is for finding the pitch and acting area of propellers; the column 
marked W contains the angle of the propeller blades, as marked on the 
drawing. 

To Find tlie Pitch. 

RULE. Multiply the diameter of the propeller by the tabular coefficient in 
the pitch column opposite the given angle, and the product is the pitch of the 
propeller. 

Example. The diameter of a propeller is 12 feet, the angle W= 60°; diameter 
of the hub 1-5 feet, and the angle w — 10P. 

{ pitch at the periphery ? 
pitch at the hub ? 
pitch at the centre of effort ? 

Pitch at the periphery = 12XH814 = 2U768 feet. 

“ “ hub = 1-5X10-97 = 16-455 feet. 

Let d and p be diameter and pitch of the hub. 

P = pitch at the centre of pressure. We shall have, 


and 


(P _ p) • (P — p) = (B - d): (0-725 D — d). 

9 


(P — p)(0-725 D — d) 
"P+ - I)-- d — -> 


p = 16-455+ 


(21-768 


16-455)(Q-725X12 —1-5) = 
12 — 1-5 


= 20-097 feet. 


To Find the Acting Area. 

RULE. Multiply the square of the diameter of the propeller by the tabular 
coefficient in the column O opposite the given angle, and the product is the 
acting area of the propeller. 

Example. The diameter of a propeller D = 13 feet, 3 inches, and the angle 
W = 60°. Required the acting area ? 

O = 13-25*X0‘679 = 119-2 square feet. 


21 * 












246 


Table for Propellers. 


TABLE I. 

Table for finding the Fitch and 
Propellers. D = 1 

Acting Area of 

• 

Angle. 

Pitch. 

Act. Area 

Angle. 

1 Pitch. 

Act. Area. 

W 

P 

o 

W 

P 

o 

5° 

36* 

0-068 

47° 

2-930 

0-573 

6 

30* 

0-082 

48 

2-828 

0-583 

7 

25-65 

0-095 

49 

2-730 

0-582 

8 

22-4 

0-109 

50 

2*635 

0-601 

9 

19-85 

0-123 

51 

2-545 

0-610 

10 

17-82 

0-136 

52 

2-455 

0-618 

11 

16-16 

0-150 

53 

2-370 

0-625 

12 

14-79 

0-163 

54 

2-283 

0-634 

13 

13-60 

0-176 

55 

2*200 

0-642 

14 

12-60 

0-180 

56 

2-120 

0-650 

15 

11-04 

0-203 

57 

2-040 

0-657 

16 

10-97 

0-217 

58 

1-963 

0-665 

17 

10-27 

0-229 

59 

1-888 

0-672 

18 

9-67 

0-242 

60 

1-814 

0-679 

19 

9-12 

0-255 

61 

1-740 

0-686 

20 

8-64 

0-268 

62 

1-670 

0-692 

21 

8-19 

0-281 

63 

1*600 

0*699 

22 

7-77 

0-294 

64 

1*530 

0-705 

23 

7*40 

0-306 

65 

1-465 

0-711 

24 

7-06 

0-319 

66 

1-400 

0-716 

25 

6-75 

0-331 

67 

1-333 

0-722 

26 

6-45 

0-344 

68 

1-270 

0-728 

27 

6-17 

0-356 

69 

1-205 

0-731 

28 

5-91 

0-368 

70 

1-142 

0-736 

29 

5-67 

0-380 

71 

1-114 

0-741 

30 

5*45 

0-392 

72 

1-021 

0-745 

31 

5-23 

0-404 

73 

0-960 

0-750 

32 

5-03 

0-415 

74 

0-900 

0-754 

33 

4*85 

0-427 

75 

0-842 

0-757 

34 

4-66 

0-439 

76 

0-783 

0-761 

35 

4-50 

0-450 

77 

0-725 

0-764 

36 

4-33 

0-461 

78 

0-668 

0-767 

37 

4-175 

• 0-472 

79 

0-611 

0-770 

38 

4-025 

0-483 

80 

0-555 

0-772 

39 

3-885 

0-494 

81 

0-498 

0-775 

40 

3-745 

0-504 

82 

0-442 

0-777 

41 

3-620 

0-515 

83 

0-386 

0-779 

42 

3-500 

0-523 

84 

0-331 

0-780 

43 

3-370 

0-535 

85 

0-275 

0-781 

44 

3-260 

0-545 

86 

0-220 

0-782 

45 

3-141 

0 555 

87 

0-165 

0 783 

46 

3-035 

0-564 

88 

0-110 

0-784 















Coefficient of Vessels. 


217 
—- 1 


' 

Tabic 

---—-- 

TABLE II. 

h 

for finding the Exponent and Coefficient of Vessels* 

Fall Lines. 

Hollow Lines. 

Exponent x. 

Coefficient k. 

Exponent x. 

Coefficient k. 

1 

o-ooo 


0-68 


1-71 

0-95 

0-024 


0.67 


1-77 

0-90 

0-228 


0-66 


1-84 

0-88 

0-326 


0-65 


1-90 

0-86 

0-432 


0-64 


1-96 

0-84 

6-558 


0-63 


2-00 

0-82 

0-692 


0-62 


1-97 

0-80 

0-836 


0-61 


1-93 

0-79 

0-902 


0-60 


1-88 

0-78 

0-978 


0-59 


1-82 

0.77 

1-050 


0.58 


1-77 

0-76 

1-12 


0-57 


1-72 

0*75 

1-20 


0-56 


1-67 

0*74 

1-28 


0-55 


1-61 

0-73 

1-35 


0-54 


1-55 

072 

1-43 


0-53 


1-50 

0-71 

1-51 


0-52 


1-44 

i 0-70 

1-59 


0-51 


1-38 

0-69 

1-64 


0-50 


1-32 




TABLE III. 




Table for fin<lin" 

tbe Slip and Acting Area# 0 = 1# 

Slip. 

Act. Area. 

Slip. 

Act. Area. 

Slip. 

Act. Area. 

Slip. 

Act. Area. 

S# 

o 

S# 

o 

S# 

o 

S# 

o 

per cent 


per cent. 


per cent. 


per cent. 


5 

84*85 

28 

4-150 

51 

0-927 

74 

0-208 

6 

60-35 

29 

3-S20 

62 

0-888 

75 

0-192 

7 

46-35 

30 

3-555 

53 

0-840 

76 

0-177 

8 

39-00 

31 

3-333 

54 

0-784 

77 

0-163 

9 

32-20 

32 

3-090 

55 

0-737 

78 

0 149 

10 

27-00 

33 

2-880 

56 

0-697 

v 79 

0-137 

11 

22-07 

34 

2-710 

57 

0-655 

80 

0-125 

12 

19-80 

35 

2-535 

58 

0-611 

81 

0113 

13 

17-32 

36 

2-366 

59 

0-581 

82 

0-103 

14 

15-20 

37 

2-222 

60 

0-546 

83 

0-092 

15 

13-52 

3S 

2-085 

61 

0-511 

84 

0-083 

16 

12-00 

39 

1-952 

62 

0-479 i 

85 

0-074 

17 

10-82 

40 

1-827 

63 

0-455 

86 

0 066 

18 

9-715 

41 

1-727 

64 

0-422 

87 

0-058 

19 

8-820 

42 

1-634 

65 

0-394 

88 

0-050 

20 

8-000 

43 

1-523 

66 

0-369 

89 

0-045 

21 

7-282 

44 

1-430 

67 

0-347 

90 

0-037 

22 

6-700 

45 

1-354 

68 

0-323 . 

91 

0-031 

23 

6111 

46 

1-275 

69 

0-300 

92 

0-026 

24 

5-635 

47 

1-195 

70 

0-281 

93 

0-021 

25 

5-200 

48 

1-127 

71 

0-263 

94 

0-017 

26 

4-785 

49 

1-068 

72 

0-241 

95 

0-012 

27 

4-444 

50 1 

1-000 

73 

1-225 



J 


. 1 

_ 

























Paddle-Wheels, 


248 


PADDLE-WHEELS. 

Letters denote. 

a — area of all the floats immersed, square feet. 

FT= the immersed angle of the paddle-wheels, at the centre of pressure. 

J{ == radius from the centre of motioy. to the centre of pressure of floats. 

0 = acting area, square feet. 

Q = a cos.fTK 

Example. Let the area of all the immersed paddles be a = 120 square feet, 
the angle W== 60°, and }X60 = 20°. Required the acting area.. 

O = 120Xcos.20° = 112 - 76 square feet.. 


Vessels* Resistance to* 

Letters denote. 

I = length in the loadline of the yessel, in feet. 

-ST - - greatest immersed section, in square feet, 

$ = area of resistance, square feet. 

. Q = displacement of the vessel in tons. 

H — horse-power required to propel the vessel M statute miles per hour. 
& = slip of the propeller or paddle-wheels. 

7c = coefficient of the vessel. 

F = resistance of the vessel, iu pounds. 


i 


35 Q 

X = --- •» m 

Ml 

1, 

log.i^ 0"63346-flog. $ +175 log .M, 

5, 

i 

II 

S3 

t* 

2, 

log .H= log. $ +2-75 log.J/-A-94148 } 

6,1 

7 i 

F = 4-3 ® iWT 2 , 

3, 

s- - 

V ® 3 +s/0 8 

7, 

IT ® M 8 

4, 

, „ log.H+1-94448 — log. ® 

log.Af — m 

8, 


The formulas 3 and 4, give the resistance and horse power too large, owing to 
the square and cube of M\ but the formulas 5 and 6 give it as correct as may be 
desired in practice. 

Example. Find the area of resistance of a vessel with dimensions l = 230 feet. 
= 490 square feet, Q — 1900 tons ? 

Exponent x = = 0*59. Find this number 

in the column x, Table II V m the next column k = 1-82, which inserted in the 
formula 2 will give the area of resistance. 

® = 490 » / -——-—■ = 35 square feet. 

V/ 490+1-82 X230* 































Tonnage of Vessels. 


249 


Speed of Steamboats* 


Screw Propellers. 


Paddle-Wheels. 

- 

1, 

M - Rnc ™« W (l — S), 

88 M 

2, 

UM 

P( i — sy 

Bcos.hW(l — S)’ 

i 88 M 

3, 

* _ i 

Pn ’ 

nRcos.hW 7 


n = 


5, 

6 , 


Example. The pitch of a propeller is P = 30 feet, and makes 40 revolutions 
per minute ; the slip is S = 0*4. Required the speed of the vessel ? 

M — — 0-4)=8.18 miles per hour. 

Explanation of Table III* 

To find the Slip. 

RULE. Divide the acting area of the propeller, or paddle-wheels by the area 
of resistance of the vessel; find the quotient in the columns of acting area , and 
opposite, in the slip column, is the slip per cent. 




TONNAGE OP VESSELS. 

The United States Custom House measurement of tonnage of vessels is 
expressed by the formula. 

Letters Denote , 

T — tonnage of the vessel. 

h = extreme beam in feet, taken above the main-wales. 

d == depth of the vessel in feet. In double-decked vessels half the beam b is 
taken as the depth. For single decked vessels, the depth is taken from the 
under side of the deck plank to the ceiling of the hold. 

I = length of the vessel in feet, from the fore-part of the steam to the after part of 
the stern-post, measured on the upper deck. 

Example. The dimensions of a vessel are l — 186 feet, b = 3Q, and <2 = 15 
feet, for a double-decked vessel. Required her tonnage ? 


T = 


30X15 

95 


(lS6-f 30^ 


795-77 tons. 


To Approximate tbe Tonnage of tbe Displacement* 

n — / °‘015 very sharp vessels. 

V JVt'X 10-022 very full vessels. 






















250 


Propeller Steamer. 


SAN JACINTO, 


United S.ates Propeller Steamer of War# 

Dimensions. 

I = 210 feet long in the load line. 

M = 477 square feet, greatest immersed section when loaded to 16 feet. 

9 inches draft of water, and Q =2150 tons, displacement. 

Required her exponent x = ? coefficient k = ? and area of resistance $ = ? 

35 X 2150 


Formula 1. page 248, Exponent x — 


= 0-75. 


477 X 210 

Find this exponent in table II. page 247, and opposite the coefficient k = 1'2. 


Formula 2. $ =477 


"Vti 


= 45 square feet. 


7"+ i'2 X 210 3 

Steam Engine of the San Jacinto# 

I am indebted to B. H. Bartol, Esqr. for the following dimensions of the 
Engine and Propeller for the San Jacinto, which was in progress at the South¬ 
wark Foundry (Merrick & Sons,) Philadelphia, when this book went to press. 
Two horizontal Engines to gear the propeller 2‘25 turns to 1. 

D — 15 feet, diameter 


d = 70 inches, diameter 
s = 4 feet, stroke of piston 


of Cylinders. 


of the 
Propeller, 


P — 22’.5 feet, pitch 
L = 3 feet, length 
|^| = 195 5 square feet, fire-grate in three boilers.. 

Cut off the steam at about £ the stroke. 

p = 15 pounds pressure per square inch in the boilers, excluding the atmosphere, 
.j. = 2*25 proportion of gearing. 

From these dimensions I will make a calculation, anticipating the perform¬ 
ance of the San Jacinto when finished. 

Calculation# 

Required the acting area of the propeller 0 = 1 
P 22-5 

_= ,—’ = l - 5, find this number in the Pitch column, Table I., page 246; 

D 15 

in the next columns W = 64° nearly, and Q = 0'7 X 15 3 
Required the slip S = ? 

o _ _157-5 

$ 45 

page 247, op posite, in the slip column S == 30 per cent nearly. 

7}s n 3 / \ 

II = T ■ ( LS cosTF-j-0 - ll 1 

480000 V ) 

2aspn 


157’5 square feet. 

3-5, find this number in the column of acting area Table III. 

cent nearly. 

- - form. 11, page 244. 


H = 


form. 22, page 149. 


24250-| 

From these two formulas we obtain the number of resolutions of the 
propeller per minute, 

Fhd 
D 


n = 




p s 


|- D(Jj S cos W+ 0-11) 


62 revo- 


Assume the effectual pressure per square inch on the piston, including the 
vacuum to be p = 25 pounds, we shall have 

5-5 X 70 
n ~ 15 

lutions per minute, or n = 

Formula 1. page 249. 
or about 9 - 6 sea miles. 


. / 25 X 4 

\/ 2-25 X 15(3 X 0-3 X cos64° + Oil) 

= 12-2 fiV 

22*5 v 62 

M =—-(1 — 0 - 30) = 11 miles per hour, 

OO 


L 























Propeller Frigate. 


251 


WABASH. 

17, S« Propeller Steam Frigate* 

Wabash is one of the six steam frigates authorized by Congress, 1854, to be 
constructed for the U. S. Navy. 

Dimensions. 

I = 271 feet, length in the load line. Beam 54 feet 4 inches. 
gT = 860 square feet, greatest immersed section. 

Q = 4700 tons tonnage of displacement. 

Propeller, true screw with two blades. 

D = 17 feet 4 inches, diameter a 
P = 23 feet, pitch K>f the propeller. 

L = 3 feet 6 inches, length * 

Two engines, applied direct on the propeller shaft. 

d = 72 inches diameter v 

s = 3 feet stroke of piston >steam> cylinders, 

p = 20 pounds effectual pressure J 

Four boilers, with vertical tubes. 

ra = 330 square feet total fire grate in the boilers, with 11500 square feet of 
heating surface. 

Steam pressure about 15 pounds per square inch in the boilers, eut off the 
steam at y the stroke. 

From these dimensions, for which I am indebted to B. H. Bartol, Esq., I will 
make a calculation of the frigate’s performance under steam. 

The calculations for the San Jacinto on the preceding page, agrees with her 
actual performance in smooth water, as near as can be observed. 


Calculation, 

Required the area of resistance, ($--=? 

35X4700 


Example 1 

Formula l r page 248. Exponent x = 

Coefficient Jc = 1*59 table II. page 247. 

For. 2, p. 248. Area of resistance $ =860 


800X271 


=0'705. 


860 


=73'6 

sq. ft. 


1-3 


27 ; 


860+l'59X271 a 

Example 2. Required the acting area of the propeller, O — ? 

P = 23 
1 ) ~~ 17553 

Find this number in the' pitch column, table' I., page 246, in the next column 
W = 67° angle of the propeller blades at the periphery, and the acting area 
O = 0-722X17’33*=217 square feet. 

Example 3. Required the slip of the propeller, S=l 

0 = 217 
® 7 3'6 

Find this number in the column O table III. page 24T, opposite in the slip 
column N = 32 per cent slip of the propeller. 


= 2'94 







252 


'sopeller Frigate. 


Example. 4. Required the number of revolutions of the propeller with 20 
pounds effectual pressure per square inch on the cylinder pistons, n = ? 


Revolu , sn= 


5-5X72 


17-33 


sj 17-33(3- 


20X3 


-5X0-32Xcost>7°i:0-ll) 


=57ipr.mt, 


Or number of revolutions n — 12*85}/ p. 

Examine 5. Required the speed of the frigate under 57& revolutions and 32 
per cent 6lip of the propeller, M — ? 

_ 23X57-5 


Speed M — 


88 


(1—0*32)=10i statute miles. 


Or 8*85 sea miles (knot) per hour 

Example 6. Required the horse-power of the engines under the above cir¬ 
cumstances, H—1 

rr oo 1AQ -D TT_2X4071'5X3X20X57-5 -, KO , 

Form. 22, p. 149. Power H— -——- =1158 horses 

24250 

Example 7. Required the horse-power of the fire-grate r— ( = ? with 15 
pounds steam-pressure in the boilers, cut off at one-third the stroke and vacuum 
in the condensor 12 pounds. 

Form. 4,page254. g . 330X883vT(l S+ 0-8X12) _ 1600 ^ 

O 1-tUU 

By this calculation the power of the fire-grate exceeds the power of the en¬ 
gine in the preceding example by 1600—1158=442 horses; consequently the 
steam can be kept higher than 15 pounds per square inch in the boilers, and 
the propeller accordingly making more revolutions per minute. 

Example 8. Assum the revolutions to n = 65 per minute, and find to what 
pressure the steam can be kept? Cut off at 12 inches of the stroke in two 
cylinders makes S = 24 inches. 


k 


8 D'Sn 
4‘66 w 


72’X24 v .65- 

4-66X7X330 


= 756, 


Find this number (nearest 748) in the Jc column, page 233, opposite in the 
last column is the required pressure, 21 pounds per square inch. 

65 revolutions == 10 knots per hour. 

Example 9. Required the consumption of coal per hour in the four boilers 
with 21 pounds of steam cut off at £ the stroke and 65 revolutions per minute 
C —l The coal to evaporate w = 7 pounds of water and k — 748 at 21 pounds. 


Form. 5, p. 255. Coal C 


2X3X72 a X12X65 

7X748 


= 4633 pounds, 

or 2 tons pr. hour. 


The Wabash is now underbuilding at the Philadelphia Navy Yard, machinery 
and propeller constructed by Merrick and Sons, Philadelphia. She will be 
ready under steam early in the spring, 1856. 




















Fresh Water Condensor. 


253 


FRESH WATER CONDENSOR. 

Fig. 2. 





OOOOOO 0-0 

oooooooo 

OOOOOOOO 

OOOOTOOO 

oooooooo 
oooooooo 
op OOOOOO 
oooooooo 

OOOOOO OQ 


Fig. 1, is a longitudinal, and Fig. 2, a transverse section of a fresh water con- 
densor with horizontal tubes. 

A, air-pump, a, fresh water. 2T, exhaust-pipe, b, hot well. T, tubes, 
c, injection pipe, d , strainer. 

The tubes are of copper one inch outside diameter, thickness of metal, 
No. 22 or 2+ wire guage, weighs b~ ounces per foot. The space occupied by 
the tubes should be about cubical, that is, the sides of the tube-plate should be 
about the length of the tubes. 

Between the injection and the tubes is a horizontal strainer, to spread the cold 
water uniformly over the tubes. Steam inside the tubes. 

Letters Denote. 

A = condensing area of all the tubes in square inches. 

I — length of tubes, height and breadth of t*ube-plate in inches. 

jV— number of tubes in the eondensor. 

Z>= diameter of steam cylinder. S = stroke of piston, iu inches. 
n — number of revolutions of engine per miuute. 


T - temperature of exhaust steam \ gee gteam tablfl| page 232o 


Jc = voluum coefficient of steam 
- D *‘V»?94 0 -|—i 


* 


0*853 y'A, 


N= 0*5128 Ip 


1-61 P. 


6*5 k 

Example. A fresh water condensor is to be constructed for an engine of 
D — 02 in. S= 70 iu. making n — 34 revolutions per minute ; T** 230°, 1225. 

Required the condensing area of tubes, A = ? 

, 62*X7GX]/3 T 


^940+230°) =295.709 square inches. 


5*5X1225 

Required the length of tubes, and sides of tube plate, l =* ? 

I = 0.853^295,7 69 = 57 inches nearly. 

Required the number of tubes in the condensor, N —1 
N= 0*512SX57 2 == 1666 tubes. 

Number of tubes in the top row 8 * de row ^ tubes. 

Should the location for. 


The tubes to be placed zigzag, as shown in Fig. 2 
the condensor not permit the cubical form, we have, 

Length tubes l = YpioJi Breadth of tube plate b = Hei S^ h = j. 61 j j 

Fres-h water produced G = gallons per minute, or about 75 per cent, 

10-1 k 0 f the feed water. 

Temperature in the hot well about 11.0° to 115°. 

„ of fresh water 120° to 130°. _ 

For fresh water condensors the capacity of the air pump should be about j 
10 per cent larger than by the rules on page 236. 


22 
























































254 


Steam boilers. 


STEAM BOILERS. 


The accompanying proportions are averages of a great number of good 
marine boilers. 

Letters denote. 

D = diameter of the steam-cylinder in inches. 

S — stroke of piston under which steam is fully admitted, in inches. 
n = number of double strokes, or revolutions per minute. 
tv = pounds of water evaporated per pound of coal, per hour. 
k = volum coefficient from the steam table. 

= fire grate in square feet, for each cylinder, and with natural draft. 

To Find tlie Area of Fire Grate* 


LH3 


D* Sn 
4 ‘66 wk 


n = 


4’66 w 7c!: 1 


S 


l, 2 . 


Example 1. A steam engine of D — 54 inches diameter of the cylinder, and 
stroke of piston 96 inches, cut off at ■§, <S=48 inches; is to make 22 revolutions 
per minute. Anthracite coal to be used, that evaporates w = 7 pounds of 
water per pound of coal, and to carry 27 pounds of steam per square inch, 
k = 649. Required the area of fire grate EEr3 = ? in square feet. 


,- E , = 54 a X48X22 


4-66X7X649 


= 145*34 square feet. 


'Example 2. A steamboiler of 1=8 == 128 square feet, is to be used for an 
engine of D = 36 inches diameter, and 64 inches stroke,—cut off the steam at 
f then aS' = 42-66 inches. Steam pressure to be kept at 25 pounds per square 
inch k = 679. w — 6-5. Required for how many revolutions per minutes can 
the steam be kept at 25 pounds ? 


n .= 


4-66^6-5^679X1 °8 
36 2 X42-66 


47'6 revolutions. 


Horse Power of tire Fire Grate. 

H = horse power of the fire grate. 

P = pressure in the boiler in pounds per square inch, excluding the 
atmosphere. 

p =a vacuum in the condensor in pounds per square inch. 

Hx Tr =^kw(P+ O.S p). 0 

"kw(P40.$p)’ " * ■ 


f i the stroke, x — 27700. saves 
x — 31400. ,, 
x = 38400. „ 

x = 45500. „ 

J „ „ * = 49100. „ 

Steam admitted throughout the stroke x => 61700. ,, 


Cut off the 
steam at 


T 55 
1 

3 55 
2 


55 
49 
38 
26 
20 

0 per cent. 


per cent 
of fuel. 


Example 3. Steamboilers are to be constructed for an engine of 650 horses, 
the steam to be cut off at 4 the stroke, a.nd P = 36 pounds per square inch, 
k = 544, to = 7‘5 pounds of water evaporated per pound of coal. Required the 
fire grate in the boilers E~l = ? in square feet. 


650 X 38400 

• 554 x 7.5 (36 + 0.8 X 11) 


136 square feet. 




















Steam boilers. 


255 


Example. 4 . Required the horse-power of a fire grate l =i = 112 square feet, 
to carry 18 pounds steam, and cut off at f the stroke ? k = 810, w = 7 pounds. 




112X18X810X7 
45500 ~~ 


=251'2 horses. 




Consumption of Coal. 

C — coal consumed in pounds per hour. 

3 D* S n 


w k 


C- 


14 H x 


k w (Pi-0‘8j9) 


Example 5 A steam engine of D — 42 inches diameter, and 48 inches stroke, 
cut off the steam at | S = 16 inches, is to make n = 65 revolutions per minute 
with a pressure of 34 pounds per square inch, k = 564, and iv = 6 pounds. Re¬ 
quired the consumption of coal in pounds per hour C — ? 


C = 


3X42 3 X16X65 

6X504 


= 1625 pounds per hour. 


Example 6. A pair of steam engines of IT = 260 horses are to be worked with 
P = 28 pounds per square inch, cut off at f the stroke, k = 635, the coal to 
evaporate w = 6'5 pounds of water per pound of coal. Required the consump¬ 
tion of coal in pounds per hour C ~ ? 


I 


c = 


14X260X31400 ^ , 

- = ll'o pounds per hour. 


630X6.5(28+0.8X10) 

It will be observed in the formulas 4 and 6, that the higher steam used, the 
less fuel and fire grate is required for the same power,—the proportion of fuel 
will be nearly as the square root of the steam pressure; and still more fuel is 
saved by cutting off the steam at an early part of the stroke. 

Fii’e Surface. 

1. In the common single returning flue boilers, the whole amount of fire 
surface from the grate to the water line should be about 25 ■—M . 

2. In flue and returning tubular boilers the whole amount of fire surface 
30 E=I- 

3. Boilers with vertical flues, (2 inches diameter,) fire outside and water in¬ 
side, fire surface 35 

Area of Flues. (Calorimeter.) 

In the common single returning flue boilers, the area of the first row should 
be about 0.18 1=1 . 

Returning row, (flue or tube) 0.13 BEZ3I 

Area of chimneys 0.16 B- —L 

Evaporative Power* Weight and Bulk of Fuel. 


Names of Fuel 

Water evap’rat’d 
per pound of 
coal. W. 

Weight per 
cubic foot. 

Cubic feet to 
stow a ton. 

Bitumenous or Anthracite coal 

6 to 9 

50 to 54 

40 to 44 

Coke. Natural Virginia 

8 to 9 

48 

48 

„ Cumberland 

8 to 10 

32 

70 

Wood, Dry Pine - - - 

4 to 5 

20 to 25 1 

100 


J 

























250 


St am boilers. 


I.iip Welded Iron Tubes. 


Outside 

Diameter. 

Standard 
Thickness 
W. G. 

lbs. Weight 
per foot. 

18 

Price 
per foot. 

66. 

Price of 
each 

safe end. 

Fire 
Surface 
per foot. 

U 

14 

1-68 

$9*25 

$9*16 

0*3273 

H 

14 

1-00 

0-23 

0*16 

0 3926 

U 

13 

1*80 

o*.s.i 

0T8 

0*4581 

2 

13 

208 

0.35 

018 

0*5236 

OL 

13 

2*25 

0 39 

0*20 

0*5890 

2 ! 

12 

2-50 

043 

023 

0*6545 

21 

12 

312 

048 

0*26 

0*7200 

3 

11 

3*50 

0 53 

029 

0*7853 

3f 

11 

4*12 

0*60 

0*32 

0 8508 

si 

10 

5*15 

0*70 

0*35 

0*9163 

3 i 

10 

5-28 

0’85 

0*38 

0*9817 

4 

10 

5*54 

1*00 

0*40 

1*0472 

5 

9 

7* 

1*60 

0*45 

1*3680 

6 

8 

10* 

2*21 

0*50 

1*5708 

7 

8 

1«* 

2*80 

0*60 

1*8326 


The “ Safe End,” of extra quality and thicker iron, is welded to one or both 
ends of flues to order, which gives increased strength to the flue where it con 
nects with the tube sheet. 

Tube varying in thickness from standard gauge, made to order. 
Manufactured and fo-r sale by Morris, Tasker and Morris, Pascal iron works, 
Philadelphia. 

WEIGHT OF STEAM BOILERS. 

Thickness and Weight. of Boiler Iron. 


Parts of the Boiler. 

Lliickuess 

by 

W Gauge. 

Thickness 

in 

Inches. 

Weight 

per 

Sq. Foot. 

Boiler legs - 

1 

0*3 

12*1 

Steam chimney- 

2 

0*284 

11 4 

Fv.rnes and crowns 

3 

0*2c9 

10*4 

Shell and steam drum- 

3 

0*259 

10*4 

Flues from 13 to IS in. diameter 

4 

0*238 

9*6 

„ 10 to 13 „ 

5 

0*22 

8*85 

„ 8 to 10 ,, 

6 

0*203 

8*17 

»» 6 to 8 t> 

7 

0*18 

7*27 

Smoke pipe 

11 

0*12 

4*83 


Weight 


of Tubular Boilers* 


Coefficient for 
Weight of Boilers. 


IQ 

18 


27 

32 


Coefficient for Weight of 
Rivets and Braces. 


Fire Surface iu 
Square Feet. 


3*8 above \ 

4* below j 

Weight of Flue Boilers. 

5*5 I above' 

6* below 


2000 


1000 


Smoke pipe and grate bars excluded. 

Example 1. Required the weight of a flue boiler with 709 square feet fire 
surfaceV ako the weight of rivets and braces? 

Weight of the boiler 709X32 =s 22,688 pounds 
of which rivets. & braces 709X 6 = 4,254 ,, 

Example 2. Required the weight of a tubular boiler with 2,460 square feet 
fire surface? also the weight of rivets and braces? 

Weight of the boiler 2460X16 = 39,360 pounds 

of which rivets & braces 2460X3*6= 8,866 „ ' 










































Optics. 


257 


OPTICS. 


Optics is that branch of philosophy which treaty of the property and motion 
of light. j , 

> n * .• 

Tlirrors* 

Example. 1. Fig. 307. Before a concave mirror of r — 6 feet radius, is placed 
an object O = 1, at d = 1-75 feet from the vertex. Required the size of the 
image I = ? ’ 

r 0 r 1X6 0 . 

° r _2 - 6—2X1-75 

Example 2. Fig. 308. Before a concave mirror of r — 5 - 25 feet radius, is placed 
an object 0 = 1, at i> — 4-5 feet from the vertex. Required the size o f the in¬ 
verted image I = ? 

. • r ~ Or - 1X5-25 - , , 

image 1 — - = —-— 14 

2 D — r 2X4*5—5*25 

Example 3. Fig. 309. Before a convex mirror of r — 1‘8 feet radius, is placed 
an object 0 — 1, at D = 3 - 15 feet from the vertex. Required the size of the 1 
image 1=^1, and the distance in the mirror d — ? 

image 3^ 3 - ^ — ^=0*222 distance d = ^._^ 0 ^ - „=0*699 ft. 


2X3-15+1-8 


2X3-15+1-8 


Example 4. Fig. 310. A parabolic mirror is h = 1-31 feet high, and d = 2-15 
feet in diameter. Required the focal distance f = ? from the vertex. 

, * , p d a 2-15*Xl£ 


= 2*646 inches. 


Optical Lenses. 

Example 5. Fig. 316. A double convex lens, of erown glass, having its radii 
\r == r — 6 inches. Required its principal focal distance/ = ? 

For crown glass the index of refraction is m = 1*52. See table. 

f= —— - = 5*768 inches. 

! J 2(1-52—1) 

Microscope. 

Letters denote. 

p — magnifying power of a lens, 
m = limit of distinct vision. 

a == limit of distinct sight, which for long-sighted eyes is about 10 or 12 
inches, and near-sighted 6 to 8 inches. For common eyes take 
a = 10 inches. 

= limit distance of the object from the optical centre at distinct vision. 

Example 6. Fig. 322. Required the magnifying power of a single mieroscope 
with principal focal distance, f = 4 - 3 inches ? 

M „ a ! f 10+4-3 

Mag. power |) = 


/ 


4-3 


=3*325 times.. 


22* 























258 


Optics. 


9 

V' 

307 

Spherical Concave Mirror. 

r = radius, and / = 5 r, focal distance of the 
mirror. 

r__ p __ 

r—2 cZ r — 2 d * 

The image disappears when d —/ = 4 r. 

° 1 ^ 

Spherical Concave Mirror. 

j _ Or Dr 

2 P—r 2 D—r 

When the object is beyond the focal 
distance the image will be inverted. 

# ■ (t> 

309 

Spherical Convex Mirror. 

j __ Or Dr 

2 D+r ““ 2P+r 

P^\ 

310 

Parabolic Concave Mirror. 

f d * . h d% . 

— 

16 h 16/ 


311 

Hyperbolic Concave Mirror. 

Heat, Light, or Sound emanating from 
the foci of a hyperbola will be reflected 
diverged, from the concave surface. 


i ... - . i 

312 

Eliptic Concave Mirror. 

Emanating rays from either of the two 
foci in an elipse, will be refracted by the 
convex surface to the other foci. 






























Optics. 


259 


Astronomical Telescopes and Opera Glasses* 

Example 7. Fig. 325. The principal focal distance f = 0‘6f> inches of the 
ocular or eye-lens. F = 58 inches the principal focal distance of the objective- 
lens. Required the magnifying power of the telescope 1 = ? 


image I = 


OF 

f 


1X58 

0-65 


= 89*23 times the object. 


The telescope is to he used at the limited distance D = 13S0 feet and D = c© . 
Required the proper lengths l — ? and micrometrical motion of the ocular o- 
eye-lens? when the limit of distinct sight a = lOin. F — 58 : 12 — 4833 feet. 
/ = 0-65 :12 = 0-05416 feet. 


I = 


1380X4*833 10X0*05416 


+• 


1380—4*833 10+0*05416 


When 2> = 1380 feet, the length l : 
When D = oo, l = 4*8333 + 0*05386 - 

Micrometrical motion of eye lens 


feet. 


4*89035 
0*05386 
4*94421 
4*88719 „ 
0*05702 „ 

0*68424 inches. 

11 

~f<T nearly. 


Table of Refractive Indices* 


Substances. 

Index. 

m. 

Substances. 

Cromate of Lead 

1 Z-97 

12-50 

Quartz - 

Muriatic Acid - 

Realgar - 

2-55 

Water - 

Diamond - * 

2-45 

Ice .... 

Glass, flint 

1-57 

Hydrogen ... 

Glass, crown 

Oil of Cassia 

1-52 

Oxygen 

1 63 

Atmospheric air- 

Oil of Olives - •> 

147 





Index. 

m. 

T54 

1.40 

1.33 

1.30 

1.000138 

1.000272 

1.000294 



314 Prism. 

An emergent rays of light a a' falling upon a 
transparent medium A (say a glass prism) will he 
trausmitted through in the direction a h, and de¬ 
livered in the direction b b', parallel to a a' a". 

V — angle of incident, v — angle of refraction. 

Indix of refraction m = S7n ‘ 

sm. v 


315 Given the direction of the emergent rays a a 

angles e and r _to find the angles u and x } —or 

the direction of the rays b b'. 

cos. z= -, cos. u=mcos. (180— z—r). 

m 

x = 180 —(e+r+tt). 

When e — u, the angle x is smallest. 

An eye in b' will see the candle in the direc¬ 
tion b' b b". 





































260 , Optic?. 


316 Double Convex Lens. 
Rr 


/= + 
f- 


_the principal 

(jn—\){R+r) focal distance. 

= vr" ' -1 v * when iff-r 
2 (m —1) 

optical centre of the lens. 


7 

Plano Convex Lens. 
f= + 

The optical centre is in the convex 
surface. 


18 


Convex-concave Lens [Meniscus.) 

f= +_ Er 

J (m—\ ){R—r) 

Draw the radii ,R' a^d r' parallel to 
another.—Draw n o, then o is the 


Double Concave Lens. 
R r 

( m —l)( J B+r)’ 


320 


Plano Concave Lens, 
r 


/— 


r tn-—\ 


The optical centre is in the concave 
surface. 


Concavo-convex Lens . 

R r 

(m—l)(R—r)' 

Draw R and r' parallel to one another. 

„ Draw n o, then o is the optical centre. 

























OPTICS. 


261 



Single Microscope. 

I: 0=f-f—d, /--££, 

V-'i 1 ' *“ t £- 


Telescope. 


O F 

f ’ 


* D—F-'a + f V a+ /' 

* + for astronomical telescope, — for opera-glasses. 


326 


Opera Glass. 


^ T t 

, * i 

Formulas are the same as for Astronomical Telescope. 


323 


When the object 0 is beyond the focal 
distance the image I will be inverted. 


I-.0~f-.D-f, I 


-°1 
d-f’ 


d ilf 





















2u2 


Geogeapsy, 


GEOGRAPHY. 



The Earth on which we live, is a round hall or sphere, with a mean diameter 
oi 7914 statute miles. The whole surface of the earth is 196800000 square miles, 
cf which only one fourth or nearly 50,000000 square miles is land, and about 
150,000000 square miles water. 

Tahle of Area and Population of the whole Earth* 


Divisions of the Earth. 

Area in Square 
Miles. 
14,491,000 
3,760,000 
16,313,000 
10,936,000 
4,500,000 

Population. 

Propm-tion to 
Square Mile. 

4 

69 

28 

5 

5 

America, ------- 

Europe, - -- -- -- - 

Asia,. 

Africa, - -- -- -- - 
Oceanica, ------- 

54,677,000 
258,354,000 
455,562,000 
61,604,000 
23,261,000 

Total,. 

50,000,000 

853,458,000 

17 

. 

About th of the whole population are born every year, and war’v rn equal 
number die in the same time; making,about one born and one dead per 


second. 

The Earth is not a perfect sphere, it is flatted at the Poles. The following rre 
her true dimensions in svatute miles of 5280 feet. 


Diameter 


Difference 
Flatted - 


Circumference 


Dimensions of the Earth* 

( 7898 - 8809 miles at the Poles. 


-I 7911-92 
(.7924-911 

- 26-0302 

- 13-015 

( 24802-486 
I 24851-640 
(24884-22 


mean, or in 45° laJ» 
“ at the Equator. 

“ Poles and Equator. 
“ at each Pole. 


round the Poles. 
Mean, or in 45° lat. 
round the Equator. 


To Find the radius of tha Earth in any givei» Latitude 

R = 3955-96(1+ 0-00164 cos.2i), statute miles. 






























Geography.- 


£68 


. Definitions^ 

Axis of the Earth is an imaginary diameter around which the earth 
revolves. 

Poles of the Earth are the two extremities of the axis, and are called North 
and South. 

Equat or of the Earth is the great circle at equo-distances from the Poles; 
it divides the Earth into the Northern and Southern Hemispheres i 

Meridian is any great circle of the Earth drawn through the Poles; hence ■ 
the Meridian runs north and south, and are at right-angles to the Equator. 

The Equator and Meridians are divided into degrees, minutes, and seconds. 

Latitude is the degrees on a Meridian counted from the Equator. 

Longitude is the degrees on the Equator or on circles parallel with the 
Equator, counted at right angles from a Meridian. 

Parallels are circles drawn through equal Latitudes; they are parallel and 
concentric with the Equator, and at right-angles to the Meridians. 

East and West is the direction of the Equator and Parallels, or at right 
angles to north and south. Turn your face towards the south, the east is on 
the left hand, and the west on the right. 

The time in which the earth makes one revolution, is divided into 24 hours, ! 

, 360 ° 1,0 
and --= 15° per hour. 


To Reduce Longitude into Time# 

RULE. Divide the number of degrees, minutes, and seconds by 15, and the 
quotient will be the time. 

Example 1. Longitude 74° 48' 15", what is it in time ? 

4Ji the answer. 

To Find the Difference in Time between two Places* 

RULE. Divide the difference in longitude by 15, and the quotient is the dif¬ 
ference in time. 


Example 2. 
cinnati ? 


Required the difference in time 


between New York and Cin- 


Longitude of Cincinnatti 
“■ “ New York - 

Difference in longitude - 


84° 27 'W 
74 07 W 
1U° 20' 


10X60 + ^0 _ 42 minutes 20 seconds, 


the difference in time. When it is 12 o’clock in Cincinnati, it is 12/i 41/ 20" in 
New York. 


Example 3. Required the difference in time between Philadelphia and Paris? 
Longitude of Philadelphia 75° 10' W 
“ Paris - 2 20 E 

Difference in longitude 77 


° 30° divided bv 15 will be 


57i 10 m the difference in time. When it is 12 o’clock.in Philadelphia, it is bh 10 m 
o’clock in Paris. 

Example 4 A vessel sails from New York for Liverpool, after she has been at 
sea about or.e week, her difference in time with New York was found to be 
2h 7 m 45 s . Required her longitude from New York ? 

15(2A 7 45 ) = 31° 58' 15" from New York. 














264 


Navigation. 


NAVIGATION. 

To navigate a vessel upon the supposition that the earth is a level plane, on 
which the meridians are drawn north and south, parallel with each other; and 
the parallels east and west, at right-angles to the former. 

The line NS represents a meridian north and 
south; the line WE represents a parallel east 
and west. 

A ship in l sailing in the direction of l V, and 
having reached V, it is required to know her 
position to the point l. which is measured by the 
line IV, and the angle Nil'; and imagined by 
the lines l a and a V 

While the vessel is running from l to V, the 
distance is measured by the log and time; and 
the course Nil' is measured by the compass 
commonly expressed in points. 

These four quantities hear the following names. 

d ' = IV, distance from l to V in miles. 

C = Nil', course, or points from the meridian. 

tj = la, departure or diiference in longitudes, in miles. 

u — al', difference in latitudes, in miles. 

I == latitude in degrees. 

L = diiference in longitude, in degrees or time. 

Formulas for Plane Sailing* 


h = d sin.C, - 
= u tan.C, - 

; 

1 , 

2 , 

„ , n tan.C 

cos./ = ——, 

15, 

■& = 60 cos./ L, 

- 

3, 

60 L 


T) = y/ d ^—u 1 , 


4, 



u = d cos. C, - 
u = t) cot. C, - 

. 

5, 

6 , 

60cos.„ ’ 

16, 

60L cos./ 
tan. C 

m 

7, 

j^_d sin.C 

60c6s./ ’ 

17, 

U = yj (/I — n \ 

m 

8 , 

T v tan.C 

18, 

d - * , - 

sin. C 

m 

9, 

60cos./ ’ 

cos. C — -7 , - 





19, 

d = - u - 

at 

10 , 

a 


cos. C, 



. ^ * 





sin.C =— - 

20 , 

, 60Z> cos./ 

m 

11 , 

d 


sin.C 



u 


d = \/ G a i XL % 

- 

12 , 

tan.C = , 

u 

21 , 

cosU " WL- ■ 

- 

13, 

sin.C - 60i cos -', 
d 

22 , 

i d sin.C 
cosi - 601/’ 

- 

14, 

tan.C - GOi cos -' 

23, 






























Navigation. 


265 


See Table of Formulas for Plane Sailing. 

Example 1. A vessel sails east north east (6 points,) 236 miles. Required her 
departure ? and difference in latitude u ? 

Formula 1. = d sin.C' = 236Xsin.6 points = 218 miles departure, and u = d 

cos.c = 236Xcos.6 points = 90-3 miles difference in latitude. 

Example 2. A ship sails in north latitude in a course C = ESE$E — 6$ 
points, at a distance of 132 miles she made a difference in longitude of L = 3° 34'. 
What latitude is she in ? 


Fwmula 14. 


cosi = 


cfsin.fi?_132Xsin.6| 


60A 


60X3+34 


= 0-59832, 


or l = 53° 15' the latitude. 

In high latitudes and very long distances, the preceding formulas will not 
give such correct results as may be desired, because they are set up with the 
supposition that the earth is a level plane; but by the aid of spherical trigo¬ 
nometry, we are enabled to ascertain courses and distances correctly, from and 
between any known points on the earth. 

Spherical Distances* 

For the spherical formulas, letters will denote. 

I = lower latitude, in degrees from the equator. 

V = highest latitude, “ “ “ 

C = course, from the latitude l to V. 

C" = course, from “ V to l. 

d = shortest distance between l and V in degrees of the great circle. 

L = difference in longitude between l and V , in degrees, or time. 

tan.m = cot.f' cos.A. 
n — 90^1 — to. 

— l, when l and l' are on one side of the equator. 

+ l, when l is On one side and V on the other. Then 


, sin./ cos.w 

cos. a =-, 

cos.m 

sin.Z cos./' 
sin.c/ 

sin.Z cos./ 


sin. C = 


sin. C' 


1 , 


2 , 


sin .d 



Example. Required the shortest distance and course from New York to 
Liverpool? 

I = 40° 42' N. latitude 1 N y , 

74°“ W. longitude J" ew lorK * 

V = 53° 22' N. latitude \ T 

2° 52' W. longitude [ Liverpool. 

L — 71° 8' difference in longitude. 

tan .to = cot 53° 22'Xcos.71° 8' = 13° 31'. 
w = 90° — 13° 31' — 40° 42' = 35° 47'. 

_ 7 sin.53°22'Xcos.35°47' . . 

For. 1. cos.cfi =-“oWfT;- = 47° 58'. 

cos.l3°31" 

Shortest distance = 47°X60+58 = 2878 geographical miles. 
sin.71° 8'Xeos.53° 22' 


sin.(7= (Tr ,. . ,_ 0 

sm.47° ob' 

course from New York NE$E. 


= 49° 23' = 4$ points 


23 












266 


Mariners’ Compass 


• 

M 
—'— 

v :j y ‘S 

fi 

s 

64:; ||| i 

1 

1 

North. 

South. 

Points. 

Degrees 

sincC. 

Cos.c. 

tan.C. 

"1 

N. 


1 _ 

1 

a 

if 

2° 49' 

5 37 

8 2« 

•0491 

•0079 

•1544 

•9968 

•9952 

•9880 

•0492 

■0983 

•1982 

N. by E. 
arul 

N. by W. 

S. by E. ( 

and -] 

S. by W. ( 

1 

n 

n 

u 

11 15 

14 4 

16 52 

10 41 

• 1936 
•2430 
•2 01 

3368 

•9811 

•9700 

•9570 

•9416 

•1989 

•2505 

•3032 

3577 

N. N. E. 
and 

N. N. AV. 

_ 

S. S. E. , 

and J 

S. S. W. | 

o 

Qi 

2i 

2f 

22 30 

25 19 

28 7 

30 56 

•3827 

•4276 

•4713 

•5140 

•9239 

•9039 

•8820 

•8577 

•4142 

•4730 

■5.343 

•5993 

N. E. by N. 
and 

N. W. by N. 

S. E. by S. ( 

and J 

S. AV. by S. ( 

3 

3'r 

3£ 

33 45 

30 44 

39 22 

42 11 

•5555 

•5981 

•6343 

•6715 

3 314^ 

-roii 

•773! 

•7410 

•6883 

•7463 

8204 

9062 

N. E. 
and 

N. AY. 

S. E, i" 

and j 

S. W ( 

4 

4^ 

4i 

4? 

45 0 

47 49 

50 37 

53 26 

•7071 

•7410 

•7731 

•8014 

•7071 

•6715 

•6345 

•5981 

1000 

1 103 

1 218 

1 348 

N. E. by E. 
and 

N. AV. by W. 

S. E. by E. ( 
and 4 

S. AY. by A7. ( 

5 

5r 

5J 

58 15 

59 4 

61 52 
64 41 

•8314 
•8577 
•8820 
•f 039 

•5555 

•51^0 

•4713 

4276 

1-496 

1-668 

1 870 

2 114 

E. N. V. 
and 

W. N. W. 

E. S. E. f 

and 4 

AY. S. AV. ( 

6 

64 

63 

6 f 

67 30 

70 19 

73 7 

75 56 

•9239 

•9416 

•3570 

•9700 

•3827 

•3368 

2901 

•2430 

2414 

2- 795 

3 295 

3- 991 

E. by N. 
and 

W. by N. 

E. by S. - f 
and 4 

AV. by S. ( 

7 

74 

74 

7f 

78 45 

81 34 

84 22 

87 11 

•9811 

•9880 

•9952 

•9988 

1936 

1544 

•0979 

•049] 

5027 

6-744 

1114 

20 32 

East or 

1 

AVest . 

8 

90° 

1 000 

0.000 

00 



















































































Navigation. 


267 
—! 

To Find tile Distances of Objects at Sea* 



Height 

Distance 

Height 

Distance 

Height 

Distance 

Height 

Distance 

in feet. 

in miles. 

in feet. 

in miles. 

in feel. 

in miles. 

infect. 

in miles. 

0-582 

1 

11 

4-39 

30 

7-25 

200 

18-72 

1 

1-31 

12 

4-5S 

35 

7-83 

300 

22-91 

2 

1*87 

13 

4-77 

40 

8-37 

400 

26-46 

3 

2-29 

14 

4-95 

45 

8-87 

500 

29-58 

4 

2-63 

15 

5-12 

50 

9-35 

1000 

32-41 

5 

2-93 

16 

529 

60 

10-25 

2009 

59-20 

6 

3-21 

17 

5-45 

i 70 

11-07 

3000 

72-50 

7 

3-43 

18 

5-61 

80 

11-83 

4000 

83-7 

8 

3-73 

19 

5-77 

99 

12-55 

5000 

93-5 

9 

3-96 

20 

5-92 

100 

13-23 

1 mile. 

96-1 

10 

418 

25 

fi-fit 

150 

16-22 




The distance being the tangent a b in statute miles, at the elevation ac, in 
feet. 

Example. 1. The light-house at a is 100 feet above the level of the sea. 
Required the distance a b. 

Height 100 feet = 13-23 miles. 

Example 2. The flog of a ship is seen from a in d. Required the distance a, d. 
when the flag is known to be 50 feet above the level d' of the sea ? 

Height of the light 100 — 13-23 miles a, b 
Height of the flag 50 = 9'35 “ b, d, 

Distance to the ship =■ 22-5S miles a, d. 

Example 3. A steamer is seen at e, the horizon b seen in the masts is assumed 
to be 16 feet above the level e'. Required the distance to the ship ? 

Height $f tlte light 100 = 13-23 miles a b, 

The assumed height 16 = 5-29 “ e b, 

Distance to the ship = 7-31 miles a e, 

To Find tlic Distance by an Observed Angle vo 

Le.tle.rs denote. 

d = distance in statute miles (a e') to the object observed. 
t, — the tangent (a b) in statute miles, or distance to the horizon. 
v = the observed angle eae', of the horizon and the loadline of the object. 
r = radius of the earth. 


= the angle b a c. 

COS.lt’ 

t 

- - 



d = cos .(w — v) V t x — r 2 

— V cos. 2 i 

(w —v )(t 2 +r 2 ) — /. 2 . 


oc> * 


































268 


POPULATION. 


POPULATION OF COUNTRIES AND CITIES IN THE WORLD. 


J. 


Names. 

NORTH AMERICA 

U« S. of America 

New York - 
Philadelphia 
Baltimore 
Cincinnati 
New Orleans 
Boston - 
Pittsburg 
St. Louis 
Chicago - 
Buffalo - 
Louisville 
Albany - 
Providence 
Newark, N. 

Charleston 
Washington 
Rochester 
Troy - - 
Richmond 
Savannah 
San Francisco 
British America 
Montreal 
Toronto * 

Quebec - 
Halifax - 
St. John - 
Cuba 
Havana - 
Santiago 
Matanzas 
Puerto Princip 
Hayti 
Portau Prince 
St. Domingo • 
Jamaica 
Kingston - - 
Mexico 
Mexico City » 
Guadalaxara - 
La Puebla - - 
San Luis Pot.osi 
Central America 
New Guatimala - - 

SOUTH AMERICA 

Brazil 

Rio Janeiro - » - 
Bahia 

Bolivia 

La Paz .... 
Equador 

Quito. 

Plata 
Buenos Ayres 
Cordova - - - - 
Paraguay 


Year 

1854 

1850 

1853 

ii 

ii 

ii 

Si 

1850 

1853 


1850 

a 

1853 

1850 

U 

1853 


1851 

1855 

1852 

1851 

1852 

1853 
<< 

<c 

a 

u 


1853 


1854 


60,000 

51,726 

50,763 

47.500 

45.500 
42,985 
40,001 
40,000 
28,785 
27,570 
23,458 
60,000 

3,634,850 

57,715 

50,000 

42,052 

33,582 

22,745 

1,009,060 

147.360 

85,242 

81,397 

46,532 

943,000 

20,000 

10,000 

377,433 

35,000 

7,853,394 

200,000 

70,000 

50,000 

40,000 

2,146,000 

50,000 

16,000,000 

6,065,000 

400,000 

120,000 

1,030,000 

20,000 

500,000 

60,000 

820,000 

85.000 

13,000 

1,000,000 


a Names. 

Year 

. Population 

) Patagonia 


1,200.000 

3 Chili 


1,200,600 

3 Santiago .... 


80,000 

3 Valparaiso - - - 

1848 

60,000 

New Grenada 


2,363,054 

Bogota ----- 


40.000 

Peru 

1851 

2,279.085 

Lima.- 

1850 

100,000 

Venezuela 

1854 

1,419.289 

Caraccas - - - . 

1853 

63,000 

Great Britain & 



Ireland 


27.686,609 

London - - - . 

1856 

2,500,000 

Manchester ... 

1851 

401,321 

Liverpool - - 

ii 

376,065 

Glasgow ----- 

i( 

347,001 

Dublin ----- 

U 

254,850 

Edinburg .... 

a 

158,015 

Prance 


35,779,222 

Paris. 

1851 

1,053,262 

Marseilles - - - - 

1852 

192,527 

Lyons . 

a 

156.169 

Spain 

1849 

13,936,218 

Madrid. 

1850 

260^000 

Barcelona - - - - 


121,815 

Portugal 

1850 

3,471,203 

Lisbon ----- 

ii 

455',217 

Belgium 

1849 

4,359,090 

Brussels ----- 

1846 

123,874 

Holland 


3,962,290 

Amsterdam ... 

1852 

228,800 

Denmark 


2,412,926 

Copenhagen - - - 

1852 

133,140 

Hamburg Free City 


200,690 

Bremen ditto - - - 


53,156 

Sweden tfc Nor- 



way 

1850 

4,810,812 

Stockholm - - - - 

1855 

100.000 

Gottenburg - - - 

ii 

30.000 

Christiania - - - - 


26,500 

Prussia 

1856 

17,178,091 

Berlin ..... 

1852 

441,931 

Austria 

1850 

36,514,466 

Vienna ----- 

1846 

407,980 

Italy 


24,733.385 

Rome- ----- 

1856 

177,461 

Naples ----- 

1851 

416,475 

Palermo ----- 

1850 

167,222 

Turkey 


35,360,000 

Constantinople - - 


786,990 

Russia 

1851 

60,098,821 

St. Petersburg - - 

1852 

533,241 

Moscow ----- 

1840 

349,068 

Odessa ----- 

1850 

71,392 

Sevastopol .... 

1855 

40,000 ‘ 

China 


387,632,907 

Pekin. 


2,000.000 

Canton ----- 


1,000,000 





















































Latitude and Longitude of Places. 


269 


Names of Places. 
N. AMERICA AND 
WEST INDIES. 
Quebec, - 
Halifax, - 
Portland light 
Buffalo, 

Chicago, 
Newburyport light 
Boston State House 
Nantucket light , 
Newport Custom, 
New York, 
Philadelphia, 

Cape Ilenlopen, 
Cincinnati, - 
St. Louis, 
Richmond, 
Washington City, 
Baltimore, 

Cape Ilatteras, 
Charleston light, 
Savannah, 

Cape Florida light, 
Pensacola, 

Mobile, 

New Orleans, 

Porto Rico, 

Cape Ilayti’s City, 
Havana, - 
Yera Cruz, • 
Mexico, - 
Porto Bello, - 
Porto Cabello, 

Cape St. August’e. 
Rio Janeiro, - 
Buenos Ayres, 
Cape Horn, - 
Valparaiso Fort, 
Panama Ft. N.E., 
San Francisco, 

ENGLAND. 

London, 

Liverpool, 

Greenwich, 

Glasgow, 

Dublin, 

Ednburgh, 

Bristol, - 
Dover, - 

FRANCE. 

Paris Observatory 
Havre de Grace, 
Cherbourgh, - 
Marseilles Observ. 
Antwerp, 

Calais, - * 

ITALY. 
Florence, 

Leghorn, 

Rome, St. Peter's, 
Naples, light, - 
A u con a, light,- 


Latitude. 


46° 49'iV 
H 38 „ 
13 36 „ 
12 53 „ 
12 0„ 
12 48 „ 
12 21 „ 
11 23 „ 
41 29 „ 
40 42 „ 
39 57 „ 

38 46 „ 

39 6 ,. 
33 36 „ 

37 32 „ 

38 53 „ 

39 18 „ 
35 14 „ 
32 42 „ 

32 5 „ 
25 41 „ 
30 24 

30 42 „ 
29 57 „ 

18 29 „ 

19 46 „ 

23 9 „ 

19 12 „ 
19 26 „ 

34 „ 
10 28 „ 

S 21 S 
22 56 S 

31 36 „ 
55 59 „ 

33 2 N 

8 57 „ 

37 47 „ 


Longitude. 


51 

31 

99 

0 

6 

53 

22 

59 

2 

52 

51 

29 

59 

0 

0 

55 

52 


4 

16 

53 

23 

95 

6 

20 

55 

57 

59 

3 

12 

51 

27 

95 

2 

35 

51 

8 

59 

1 

19 

IS 

50 

55 

2 

20 

19 

29 

55 

0 

6 

19 

38 

55 

1 

37 

13 

18 

95 

5 

22 

51 

13 

55 

4 

21 

50 

68 

95 

1 

51 

13° 

46 

55 

11 

16 

13 

32 

*9 

10 

IS 

11 

54 

59 

12 

27 

to 

50 

55 

14 

16 

13 

33 

99 

13 

30 


71 16 

63 35 
70 12 

78 55 
87 35 

70 49 

71 4 

70 3 

71 19 

74 00-7 

75 10 

75 4 
84 27 

89 36 
77 27 
77 0-3 

76 37 
75 30 

79 54 
81 8 

80 5 
S7 10 

57 59 

90 0 

66 7 

72 11 
82 22 

98 9 

99 5 
79 40 
68 7 
34 57 
43 9 

58 22 

67 16 
71 41 
79 31 

122 21 


0 

W\ 

>' 

jj 

59 

E\ 


” 

55 


Names of Places. 

GERMANY. 
Berlin, - 
Bern, 

Rotterdam, 
Antwerp, 
Amsterdam, 
Bremen, 

Hague, - 
Hamburg, 
Lubeck, - 

AUSTRIA 

Vienna, • 

Venice, - 
Trieste castle, 

TURKEY, 
Ragusa, mole, 
Athens Pkilopa., 
Salonica, 
Constantinople, 
SWEDEN AND 
NORWAY. 
Stockholm, 
Gothenburg, - 
Christiania, - 
Bergen. - 
Wieby Gotland, 
DENMARK. 
Copenhagen, • 
Elsineur, 

RUSSIA. 

St. Petersburg, 
Moscow, - 
Revel, - 
Riga, 

Cronstadt, 

Abo, 

Odessa, - 

SPAIN. 


Latitude. Longitude. 


52 

46 

51 

51 

52 

53 

52 

53 
53 


31 N 
57 

54 “ 
13 “ 
22 “ 
*5 “ 

4 “ 

o3 “ 

52 “ 


13 “ 
26 “ 
39 “ 


42 3S tt 

37 58 u 

10 39 tt 

11 1 tt 


59 
57 
59. 

60 
57 


55 

56 


21 “ 
42 a 
55 “ 

21 tt 

39 ‘t 

41 ft 
2 “ 


59 

55 
59 

56 

59 

60 
±6 


56 tt 
46 tt 

26 tt 

51 tt 
58 tt 

27 ft 

27 tt 


13 24 
7 25 
4 28 
4 24 

4 51 

5 49 
4 16 
9 56 

10 49 


16 23 
12 21 
13 43 


18 7 
23 44 
22 57 
28 59 


18 4 

11 57 
10 52 

5 20 
18 17 

12 34 
12 37 


23 * 


Madrid, - 

10 25 ft 

3 42 W 

Barcelona, 

11 23 ft 

211 E 

Algiers light, - 

36 49 f 

3 1“ 

Gibraltar, 

36 6 tt 

5 20 IF 

Carthagena ob3er., 

37 36 ft 

11“ 

PORTUGAL. 



Oporto, - 

11 11 ft 

8 38 “ 

Lisbon, - 

38 42 “ 

9 9“ 

Cape St. Vincent, 

37 3 “ 

9 2“ 

SICILY. 



Messina, 

33 12 “ 

15 35 E 

Palermo, 

18 S “ 

13 22 “ 

Malta, - 

35 51 « 

14 13 “ 

CHINA. 



Peking, - 

39 54 “ 

116 28 “ 

Canton, 4 

23 7 “ 

113 14 « 

Cape of Good Hope 

34 22 S 

18 30 “ 

Sidney, Australia, 

34 0 

151 23 “ 

Jerusalem. Piles., j 

31 48 N 

37 20 “ 


30 19 

35 33 
24 46 
23 57 

29 51 
22 15 

30 42 































Distances by Sea. 


c 0 

o ^ 

£ oo 

P o 

ci 

o o" 
o 

CO 

•H 

O 

P 

cS 

Ph 

a 

c3 

GG 


o +o fO 

CO 'o 'oo 

X- H 05 

O) N o 


05 05 05 

N ^ O 


O 

O 


to to to 


M !N N (N CO 

co co to to 03 

^ rj< ^ if ^ 


o +o fC 
O 'o ICO 
rH CO CO 
CO CO CO 


eS 

a 

» 

o 

Q 

a 

a 

% 


05 

tO 


I— 03 


o co x^ x- 


vOtOtOHCOCOCOCOrH 


*® *® *1—1 *® 

o 

o 

o 

03 

CO 

05 

o 

CO 

CO 

03 

o 

o 

O 

o 

CO 

tO 

CO 

CO 

CO 

H 

to 

x^ 

03 

H 

r-H 

rH 

rH 

rH 




+ H 

, o 

CO 

IO • 

fo 

o 

o 

To 

' H 

1 tO ' 

**co 

1 CO 

CO 

CO 

03 

CO 

CO 

CO 

X- 

CO 

H 

rH 

03 

CO 

H 

05 

X— 

CO 


• 

c? 

CO 


cs 

X^ 

x-- 

w 

CO 


w> 

o 

w 

W' 

Ov 


o 

o 

C5 

p 

03 

o 

co 

CO 

co 

Hf 

05 

co 

rH 

CO 


CO 

CO 

X- 

CO 

o 

o 

CO 

o 

CO 


CO 

CO 

rH 

o 

co 

CO 

CO 

to 

co 

H 

H 

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27 4 


ASTKOJfOMT. 


ASTRONOMY. 


Asthonomv is that branch of Philosophy which treats of the properties of 
heavenly bodies. 

The mean Solar day is divided into 24 hours. 

* A Sidereal year is ^ - 3C5d Ca 9™ 9*6«. 

f An Anomalistic year is - - - - - - 365<* C A 13™ 49‘3«. 

X A Tropical year i3 - - - - - - - 305 tJ C A 13™ 49*3*. 

* Sid-erea.1 year is the time in which the Earth makes one revolution round 
the Sun. in reference to a fixed star. 

f Anomalistic year is the time which the Earth occupies between each 
perihelion to the sun. 

X Tropical year is the periodical return of seasons. 

Mean distance from the Earth to the Sun 95000000 miles or 11992 diameters 
of the Earth. 

Inclination of the Ecliptic to the Equinoctial 23° 28' 40". 

The Sun subtends an angle from the Earth of 32' 3". 

Horizontal paralax of the Sun S'6" seconds. 

Velocity of a point at the Equator by the rotation of the Earth 152'5 feet per 
second. 

Cycle of (He Sun is the period of 28 years at which the days of the week 
return to the same days at the month. 


The Moon. 

Distance from the Earth to the Moon is 237000 miles, = 30 diameters of the 
Earth, = about 0 25 the diameter of the Sun, or the diameter of the Sun is twice 
the diameter of the Moon’s orbit around the Earth. 

Diameter of the Moon is 2160 miles, or about 0-2729 of the Earth’s diameter. 
Volume of the Moon compared with the Earth is 0*02024. 

Density of the Moon compared with the Earth is 0*5657. 

Mass of the Moon compared with the Earth Is 0*011399, 

Inclination of the Moon’s orbit to the Ecliptic 5° S' 48". 

The Moon subtends an angle from the Earth of 31' 7". 

Mean Sidereal revolution of the Moon 27*32166 days. 

Mean Synodical revolution of the Moon 29-5305SS7 days. 

The Moon passes the meridian in periods of 24-81.4 hours, or 48™ 5C* later 
every day. 

Moon’s Age is the number of days from the last new-moon, 

JVum’w of Months for the Moon’s Age. 


January, 

February, 

March, 


0, 

April, 

2, 

July, 

5, 

2, 

May, 

3, 

August. 

6, 

1, 

June, 

4, 

September, 



October, 8, 
November, 10, 
December, 10. 


To Find the Moon's Age on any given day. 

RULE. Add together the day of the month, Epact for the year, and the Tab¬ 
ular number for the month, the sum will he (he moon’s age. If it exceed 30, 
reject 30’s, said the remainder will be the moon’s age. 


Example. Find out if it will be moonlight at Christmas, 1855? 
Number of day = 25 in December 

Epact for 1855 (see Table) = 12 
Tabular number of December = 10 


47 — 30 = 17 the moon’s ago 


or in its third quarter, consequently moon-light at mid-night. 


Golden Number or Lunar Cycle 

is the period of 19 yeaars at which the changes of the moon fall on the same 
days of the month. 


To find the Golden number. 

RULE. Add one to the given year, divide the sum by 19, and the remainder 
will be the Golden number. 
















Almanac tor tiie 19tii Century. 


275 




Dam. 

bi 

Trs. 

Dags. 

let¬ 

a 



ter. 

a 

ISO! 

Saturd.* 

FE 

4 

1001 

Sunday. 

D 

15 

1802 

-Monday. 

C 

26 

1803 

Tuesday 

Thursd* 

B 

7 

1801 

AG 

18 

1S05 

Friday. 

F 

29 

1808 

Saturd. 

E 

11 

1807 

Sunday. 

I) 

22 

ISOS 

Tuesd.* 

CB 

3 

1809 

Wedns. 

A 

14 

1S10 

Thursd. 

G 

25 

1811 

Friday. 

F 

6 

1812 

Sunday* 

ED 

17 

1813 

Monday. 

Tuesd. 

C 

28 

1814 

. B 

9 

1815 

Wedns. 

A 

20 

1816 

Friday* . 

GF 

1 

1S17 

Saturd. 

E 

12 

1818 

Sunday. 

D 

23 

1819 

Monday. 

C 

4 

1820 

Weds.* 

BA 

15 

182 L 

Thursd. 

G 

26 

1S22 

Friday. 

F 

7 

1823 

Saturd. 

E 

IS 

1821 

Mouda*. 

DC 

29 

1825 

Tuesd. 

B 

11 

3 826 

'• edsd. 

A 

22 

1827 

Thursd. 

G 

3 

3828 

Saturd.* 

FE 

14 

1829 

Sunday. 

D 

25 

1830 

Monday. 

Tuesd. 

C 

6 

1831 

B 

17 

1832 

Thors.* 

AG 

28 

1S33 

Friday. 

F 

9 




Dom. 


Trs. 

Days. 

let¬ 




ter. 


1834 

Saturd. 

E 

20 

1835 

Sunday. 

D 

1 

1836 

Tuesd.* 

CB 

12 

1837 

Wedsd. 

A 

23 

1838 

Thursd. 

G 

4 

1839 

Friday. 

F 

15 

1840 

Sund.* 

ED 

26 

1841 

Monday. 

C 

7 

IS 42 

Ttte-sd. 

B 

18 

1843 

Wedsd. 

A 

29 

1S44 

F riday.* 

GF 

11 

1845 

Saturd. 

E 

22 

1846 

Sunday v 

D 

3 

1847 

Monday. 

C 

14 

1848 

Wedsd*. 

BA 

25 

1849 

Thursd. 

G 

6 

1850 

Friday. 

F 

17 

1851 

Saturd. 

E 

28 

1852 

Mond.* 

DC 

9 

1853 

Tuesd. 

B 

20 

1854 

Wedsd. 

A 

1 

1855 

Thursd. 

G 

12 

1856 

Saturd.* 

FE 

23 

1857 

Sunday. 

D 

4 

1858 

Monday. 

C 

15 

1859 

Tuesd. 

B 

26 

I860 

Thurs.* 

AG 

7 

1861 

Friday. 

F 

IS 

1862 

Saturd. 

E 

29 

1863 

Sunday, 

D 

11 

1864 

Tuesd*. 

CB 

22 

IS 65 

Wedsd. 

A 

3 

1866 

Thursd. 

G 

14 

1867 

Friday. 

F 

25 




Dom. 

hg 

Trs. 

Days. 

let¬ 

3 

5 



ter. 

O 

1868 

Sund* 

ED 

6 

IS 69 

Monday. 

C 

17 

1870 

Tuesd. 

B 

23 

1871 

Wedsd. 

A 

9 

1872 

Friday*. 

GF 

20 

1873 

Saturd. 

E 

1 

1874 

Sunday. 

D 

12 

1875 

Monday. 

C 

23 

1876 

tV edsd.* 

BA 

4 

1877 

Thursd. 

G 

15 

1878 

Friday. 

F 

26 

1879 

Saturd. 

E 

7 

18S0 

Mond.* 

DC 

IS 

1881 

Tuesd. 

B 

29 

1882 

W’edsd. 

A 

11 

1883 

Thurs. 

G 

22 

1884 

Saturd.* 

FE 

3 

18S5 

Sunday. 

D 

14 

1886 

Monday. 

C 

25 

1887 

Tuesd. 

B 

6 

1S88 

Thurs.* 

AG 

17 

1S89 

Friday. 

F 

28 

1890 

Saturd. 

E 

9 

1891 

Sunday. 

D 

20 

1892 

Tuesd * 

CB 

1 

1893 

Wed so; 

A 

12 

1894 

Thursd. 

G 

23 

1895 

Friday, 

F 

4 

1896 

Sund.* 

ED 

15 

1S97 

. Monday. 

C 

26 

1898 

Tuesd. 

B 

7 

1899 

Thurs. 

A 

18 

1900 

Friday*. 

GF 

29 


In Leap years ta':e .January ,* February * 


February, 

February * 


January 

January ,* 

: September 


March, 


May. 


April , 


June. 

November. 

August. 


October. 

July. 

December. 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 « 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 






Example. On what day i ; the week will the fourth of July fall in the year 

1868? , « j 

See Table 1868 — Sunday*. In tlve Table of months, July column, Sunday 
is on the 5th; consequently the fourth falls on Saturday. 

To Find the Latitude of a place hy the Meridian Altitude 

of tile Stiti* 

Letters denote. 

j — meridian altitude of the sun's centre above the horizon, in decrees and 
minutes. (At sea the sun’s lower limb is generally observed, then for correc¬ 
tions of semi-diameter dip of horizon, parallex, add 12 minutes to the observed 
altitude, and the sum will be the centre altitude very nearly.) 

D i= declination of the sun at the time of observation, to be found on 
page 278. 

I — latitude of the place of observation. 


























276 


Astronomy. 


„ f = 90 — A±D, - - 1, 

41*2 j ' I 1=90 —1 ± A ... 2, 

! < - 1 X> - + 90 ± 1 ± /, i - - 3, 

Where the quantities have double signs, plus and minus, use the upper one 
when the latitude and declination are of equal names ; and the lower one when 
the latitude and declination are of different names. 

Example. On the 25th day of October, 1853, in north latitude was observed 
the sun’s meTidian centre altitude to be A = 37° 53'; the declination on that 
day was D = 12° 10' south. Required the latitude ? 

7 = 90 — 37° 53' — 12° 10' = 39° 57' the latitude of Philadelphia. 

To Find the Time when the Sun Rises or Sets. 

Let v be the angle of time before or after six o’clock when the sun rises or 
sets; this angle divided by 15 and added to, or subtracted from six o’clock will 
be the true time when the sun rises or sets. 

sin.i; = tan.D tan./, - - - 4, 

Example. What time does the sun rise and set, on the 27th day of July, 
1854, in north latitude l — 42° 6' ? 

Sun’s declination { % = ! n mornin S‘ ) North. 

(. D — 19 5' in the evening, j 

sin.u = tan.l9° 12'Xtan.42° 6' = 0-31454 or v = 18° 20'. 

18° 20': 15 = 1* 13™ 20* subtract from 6 a 

5A 59»i go* 

Sun rises at 4* 46™ 40* in the morning, 
sin.a = tan.19° 5'Xtan.42° 6' 0-312611 or v = 18° 13'. 

6* 

18° 13' r 15 = 1 12 56 ad d to 6* 

Sun sets at l h 12 OT 56* 

To Find the Length of Day and Night* 

RULE. Double the time when the sun sets is the length of the day. 

RULE. Double the time when the sun rises is the length of the night. 

To Find the apparent Time by an Altitude of the Sun. 

Let L be the angle of time from 12 o'clock, (noon,) when the sun’s altitude a 
is observed, 

a = the observed altitude of the sun, (if the sun's lower limb is observed add 
12' for corrections), 

A and v, same as in the preceding examples. 


T sin.fl(l±sin.t;) _ . 

cos .L = -\— - t + sm.t), 

sin.A 


5 , 


The sign -f or — is to be used as before described. 

Example. On the 11th of May, 1853, the sun’s altitude in the afternoon was 
observed to be a = 42° 30'. in the latitude l — 33° 40'; the sun’s declination at 
the time of observation was D — 18° N. Required the apparent time. 

sin.v = tan.l8°xtan.33° 40' = 0*21642, - 4, 

A ='90 — 33 40 +18 = 74° 20' - - 2, 


T _ sin.42 J 30'(1+0*21642) 
sin. 74° 20' 

50° 25 


0*21642 = 0*63709, 5, 


or 50° 25' is thS angle X, — = 3a 21™ 40*,the apparent time of observation. 

If the altitude is taken in the forenoon, subtract the obtained time from 12.4 
and the remainder is the apparent time. 










Elements of the Planetary System, 


277 


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*27.8 


Sun’s Declination. 








SUN’ 

S 

DECLINATION 

• 






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Sou. 

Feb. 

Sou. 

Mar. 

Sou. 

Apr. 

Nor. 

May 

Nor. 

June 

Nor. 

July 

Nor. 

Aug. 

Nor. 

Sep. 

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Dec. 

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6 

13 

16 

21 

22 

36 

22 

46 

16 

55 

6 

42 

4 

52 

15 

48 

22 

27 

5 

6 

22 

27 

15 

29 

5 

30 

6 

36 

16 

38 

22 

42 

22 

40 

16 

38 

6 

20 

5 

15 

16 

6 

22 

34 

6 

7 

22 

19 

15 

11 

5 

7 

6 

58 

16 

55 

22 

48 

22 

34 

16 

21 

5 

57 

5 

38 

16 

24 

22 

41 

7 

8 

22 

11 

14 

52 

4 

44 

7 

21 

17 

11 

22 

53 

22 

27 

16 

4 

5 

35 

9 

1 

16 

41 

22 

47 

8 

9 

22 

3 

14 

32 

4 

20 

7 

43 

17 

27 

22 

59 

22 

20 

15 

47 

5 

12 

6 

24 

16 

59 

22 

53 

9 

10 

21 

54 

14 

13 

3 

57 

8 

5 

17 

43 

23 

3 

22 

13 

15 

29 

4 

49 

6 

47 

17 

16 

22 

58 

10 

11 

21 

A\ 

13 

53 

3 

33 

8 

27 

17 

58 

23 

7 

22 

5 

15 

12 

4 

26 

7 

10 

17 

32 

23 

3 

11 

12 

21 

35 

13 

33 

3 

10 

8 

49 

18 

13 

23 

11 

21 

56 

14 

54 

4 

3 

7 

32 

17 

48 

23 

8 

12 

13 

21 

24 

13 

13 

2 

46 

9 

11 

18 

28 

23 

15 

21 

48 

14 

35 

3 

40 

7 

55 

18 

4 

23 

12 

13 

14 

21 

14 

12 

53 

2 

22 

9 

32 

18 

43 

23 

18 

21 

39 

14 

17 

3 

17 

8 

17 

18 

20 

23 

15 

14 

15 

21 

3 

12 

32 

1 

59 

9 

54 

18 

57 

23 

20 

21 

29 

13 

58 

2 

54 

8 

39 

18 

36 

23 

18 

15 

16 

20 

51 

12 

11 

1 

35 

10 

15 

19 

11 

23 

23 

21 

20 

13 

39 

2 

31 

9 

1 

18 

51 

23 

21 

16 

17 

20 

39 

11 

50 

1 

11 

10 

36 

19 

24 

23 

24 

21 

9 

13 

20 

2 

8 

9 

23 

19 

5 

23 

23 

17 

18 

20 

27 

11 

29 

0 

48 

10 

57 

19 

37 

23 

26 

20 

59 

13 

1 

1 

45 

9 

45 

19 

20 

23 

25 

18 

19 

20 

15 

11 

8 

0 

24 

11 

18 

19 

50 

23 

27 

20 

48 

12 

41 

1 

21 

10 

7 

19 

34 

23 

26 

19 

20 

20 

2 

10 

46 

0 

ON 

11 

38 

20 

03 

23 

27 

20 

37 

12 

21 

0 

58 

10 

29 

19 

47 

23 

27 

20 

21 

19 

49 

10 

24 

0 

23 

11 

59 

20 

15 

23 

27 

20 

25 

12 

1 

0 

35 

10 

50 

20 

1 

23 

27 

21 

22 

19 

35 

10 

3 

0 

47 

12 

19 

20 

27 

23 

27 

20 

13 

11 

41 

0 

11 

11 

11 

20 

13 

23 

27 

22 

23 

19 

21 

9 

41 

1 

11 

12 

39 

20 

40 

23 

26 

20 

2 

11 

21 

012S 

11 

32 

20 

26 

23 

27 

23 

24 

19 

7 

9 

18 

1 

34 

12 

59 

20 

51 

23 

25 

19 

48 

11 

0 

0 

36 

11 

53 

20 

38 

23 

25 

24 

25 

18 

52 

8 

56 

1 

58 

13 

18 

21 

1 

23 

24 

19 

37 

10 

40 

0 

59 

12 

14 

20 

50 

23 

24 

25 

26 

18 

37 

8 

34 

2 

21 

13 

38 

21 

12 

23 

22 

19 

24 

10 

19 

1 

22 

12 

35 

21 

1 

23 

22 

26 

27 

18 

21 

8 

11 

2 

45 

13 

57 

21 

22 

23 

20 

19 

10 

9 

58 

1 

46 

12 

55 

2l 

12 

23 

19 

27 

28 

18 

6 

7 

49 

3 

8 

14 

16 

21 

32 

,23 

17 

18 

56 

9 

37 

2 

9 

13 

15 

21 

24 

23 

16 

28 

29 

17 

49 



3 

32 

14 

34 

21 

41 

23 

14 

18 

42 

9 

15 

2 

33 

13 

35 

21 

34 

23 

13 

29 

30 

17 

33 



3 

55 

14 

53 

21 

50 

23 

10 

18 

28 

8 

54 

2 

56 

13 

55 

21 

44 

23 

9 

30 

31 

17 

16 



4 

18 



21 

59 



18 

13 

8 

32 



14 

14 



23 

5 

o \ 


Declination of tlie Sun and equation of time for the years 


Leap years 1852, 

•56, 

•60, 

•64, in New York at 6h a.m. 

1853, 

•57, 

•61, 

1865, „ „ apparent noon, 

1854, 

•58, 

•62, 

&c. „ „ 6h p.m. 

1855, 

•59, 

•63, 

Ac. „ 12h midnight. 


Exampe 1. Required the sun’s declination in New York at 10 o’clock a.m. on 
the 13th of April, 1856? 

From the table April 13 declin, 9° 11' N. 

» » }t )> 14 = » 9 32 


Difference 21 
Dec. 9° 12' 

Correction 21 (10—6): 24 — 3£ add. 


Required declination 9° 15' 30" the answer. 

Note. In leap years the declination and equation of time must he taken one 
day earlier in the tables for January, and February: for instance, declination 
on the 20th of February 1856 is 7° 49'. 



























Equation of Time. 


279 


EQUATION OF TIME. 


cc 

Jan. 

Feb. 

Mar. 

Aph. 

May! 

June 

.July 

Aug. 

Sep. 

Oct. 

Nov. 

Dec. 

WJ 

c? 

Add 

Add 

Add 

Add 

Sub. 

Sub. 

Add 

Add 

Sub. 

Sub. 

Sub. 

Sub. 


« 

m. 

S. 

m. 

S. 

m. 

S. 

m. 

S. 

tu. 

s. 

m. 

s. 

m 

s. 

m. 

s. 

m. 

S. 

m. 

S. 

m. 

s. 

m. 

S. 

o 

1 

4 

4 

13 

58 

12 

31 

3 

52 

3 

4 

2 

2 s 

3 

31 

6 

0 

0 

13 

10 

24 

16 

16 

10 

37 

l 

2 

4 

33 

14 

5 

12 

20 

3 

34 

3 

11 

2 

19 

3 

42 

5 

56 

0 

32 

10 

43 

16 

17 

10 

14 

2 

3 

4 

59 

14 

11 

12 

7 

3 

17 

3 

18 

2 

9 

3 

53 

5 

52 

0 

51 

11 

2 

16 

17 

9 

50 

3 

4 

5 

27 

14 

17 

11 

54 

2 

59 

3 

24 

1 

59 

4 

4 

5 

47 

1 

11 

11 

20 

16 

16 

9 

25 

4 

5 

5 

54 

14 

22 

11 

40 

2 

41 

3 

29 

1 

49 

4 

15 

5 

41 

1 

30 

11 

37 

16 

14 

9 

0 

5 

6 

6 

20 

14 

26 

11 

26 

2 

24 

3 

34 

1 

38 

4 

25 

5 

35 

1 

50 

11 

55 

16 

11 

8 

35 

6 

7 

6 

46 

14 

29 

11 

12 

2 

7 

3 

38 

1 

27 

4 

35 

5 

28 

2 

10 

12 

12 

16 

8 

8 

9 

7 

8 

7 

12 

14 

31 

10 

57 

1 

50 

3 

42 

1 

16 

4 

44 

5 

20 

2 

31 

12 

29 

16 

4 

7 

43 

8 

9 

7 

37 

14 

33 

10 

41 

1 

32 

3 

45 

1 

4 

4 

53 

5 

12 

2 

51 

12 

45 

15 

58 

7 

16 

9 

10 

8 

1 

14 

33 

10 

26 

1 

16 

3 

48 

0 

53 

5 

2 

5 

3 

3 

12 

13 

1 

15 

52 

6 

48 

10 

11 

8 

25 

14 

33 

10 

10 

1 

0 

3 

50 

0 

41 

5 

10 

4 

54 

3 

33 

13 

16 

15 

46 

6 

21 

11 

12 

8 

48 

14 

32 

9 

53 

0 

42 

3 

51 

0 

29 

5 

17 

4 

44 

3 

54 

13 

31 

15 

38 

5 

53 

12 

13 

9 

10 

14 

31 

9 

37 

0 

26 

3 

52 

0 

16 

5 

25 

4 

34 

4 

15 

13 

45 

15 

30 

5 

24 

13 

14 

9 

32 

14 

28 

9 

20 

0 

11 

3 

53 

0 

4 

5 

31 

4 

23 

4 

36 

13 

59 

15 

20 

4 

56 

14 

15 

9 

53 

14 

25 

9 

3 

OS 2 

3 

53 

0A 9 

5 

38 

4 

11 

4 

57 

14 

12 

15 

10 

4 

27 

15 

16 

10 

14 

14 

21 

8 

45 

0 

17 

3 

52 

0 

21 

5 

43 

3 

59 

5 

18 

14 

25 

14 

59 

3 

57 

16 

17 

10 

33 

14 

16 

8 

25 

0 

31 

3 

51 

0 

34 

5 

48 

3 

46 

5 

40 

14 

37 

14 

47 

3 

28 

17 

18 

10 

52 

14 

11 

8 

10 

0 

45 

3 

49 

0 

47 

5 

53 

3 

33 

6 

1 

14 

48 

14 

34 

2 

58 

18 

19 

11 

11 

14 

5 

7 

52 

0 

58 

3 

46 

1 

0 

5 

57 

3 

20 

6 

22 

14 

59 

14 

21 

2 

28 

19 

20 

11 

28 

13 

58 

7 

33 

1 

12 

3 

44 

1 

13 

6 

1 

3 

6 

6 

43 

15 

9 

14 

6 

1 

58 

20 

21 

11 

45 

13 

51 

7 

15 

1 

24 

3 

40 

1 

26 

6 

4 

2 

51 

7 

4 

15 

20 

13 

51 

1 

28 

21 

22 

12 

1 

13 

43 

6 

57 

1 

37 

3 

36 

1 

39 

6 

6 

2 

36 

7 

25 

15 

29 

13 

35 

0 

58 

22 

23 

12 

16 

13 

35 

6 

38 

1 

48 

3 

31 

1 

52 

6 

8 

2 

21 

7 

45 

15 

37 

13 

18 

0 

28 

23 

24 

12 

30 

13 

25 

6 

20 

2 

0 

3 

26 

2 

4 

6 

10 

2 

5 

8 

6 

15 

44 

13 

0 

A0 

4 

24 

25*12 

44 

13 

16 

6 

1 

2 

10 

3 

22 

2 

17 

6 

11 

1 

49 

8 

26 

15 

51 

12 

42 

0 

32 

25 

26 

12 

57 

13 

5 

5 

43 

2 

21 

3 

15 

2 

30 

6 

11 

1 

33 

8 

47 

15 

57 

12 

22 

1 

2 

26 

27 

13 

10 

12 

55 

5 

24 

2 

30 

3 

9 

2 

42 

6 

11 

1 

16 

9 

7 

16 

2 

12 

2 

1 

32 

27 

28 

13 

21 

12 

43 

5 

6 

2 

40 

3 

2 

2 

55 

6 

10 

0 

58 

9 

26 

16 

6 

11 

42 

2 

1 

28 

29 

13 

31 



4 

47 

2 

48 

2 

54 

3 

7 

6 

8 

0 

41 

9 

46 

16 

10 

11 

20 

2 

31 

29 

30 

13 

41 



4 

29 

2 

56 

2 

46 

3 

19 

6 

6 

0 

23 

10 

5 

16 

13 

10 

58 

3 

0 

30 

31 

13 

50 



4 

11 



2 

37 



6 

4 

0 

5 



16 

15 



3 

28 

31 


Example. 2. Required the sun’s declination in San Francisco at 4h 46m p.m., 
on the 5th of Aug. 1855. 

From the table Aug. 5 declin. 16° 55' N. 
a » f> a ® n 38 


Difference 17 

Long, of San Francisco 122° 

„ New York 74 

48:15 = 3h 12m difference in time. 
Time in S. F 4 46 

Midnight 12h — 8h = 4h X 17 ; 24 = 3' nearly. 

16° 45' 

Correction add 3 


Required declination 16° 48' the answer. 

Example 3. On the 5th day of Nov. 1857, at 2h 21m 56s apparent time p.m. 
Required the mean time ? 

Apparent time 2h 21m 56s 

Equation of time, sub. 16 14 


Mean time 2h 5m 42s the answer. 

Note. Add the equation of time to or subtract from the apparent time, is 
the mean time. 








































280 


Moon’s Age 



EPACT OF THE YEAR. 

d. h. 

d.h. 

d. h. 

d. h. 

d. h. 

d. h. 

d. h. 

d h. 

d. h 

d. h. 

d. h. 

d. h. 

1850 

1851 

1852 

1853 

1854 

1855 

1856 

1857 

1858 

1859 

I860 

1861 

17 17 

28 8 

910 

20 1 

1 3 

11 18 

2310 

4 12 

15 3 

25 17 

7 21 

1812 

1862 

1863 

1864 

1865 

1866 

1S67 

1868 

1869 

1870 

1871 

1872 

1873 

29 3 

10 6 

21 21 

2 23 

1315 

24 6 

6 8 

16 23 

2718 

8 20 

1915 

0 17 

EPACT OF THE MONTH. 


Jan. 

Feb. 

Mar. 

Apr. 

Vay 

June 

July 

Aug. 

Sep. 

Oct. 

Nov. 

Dec. 

0 0 

1 17 

0 4 

1 16 

2 3 

. 14 

4 2 

5 13 

7 

7 11 

8 23 

9 10 


To the time of high water, or the time when the moon passes the meridian, 
add two minutes for every exceeding hour of the moon’s age, for corrections. 


MOON’S POSITION. 


HIGH WATER IN DIFFERENT PLACES. 


p 

8 


Half 

to 

Pi 

P 

P 


Full 


os 

a. 

>o 

s 

P 


H 

Half 

#>■ 

c+- 

O' 

»p 

P 

P 


Face 


[oon. 

High water 
in N. York, 
h. m. 

Add to, or subtract from the 
time of high water in N. Y. 
is the time of high water in 
the desired place. 

Rise 

Ft. 

Age 

d. 

South, 
h. m. 

0 

12 0p.m. 

8 

37 a.m. 

New York 


oh 0m 

5 

1 

12 49 

99 

9 

21 

99 

Quebec 

add 

8 49 

17 

2 

1 38 

99 

10 

2 

99 

Halifax 

sub. 

1 0 

8 

3 

2 26 

99 

10 

40 

99 

Boston 

add 

3 54 

12 

4 

3 26 


11 

16 


New Haven 

add 

2 32 

17 

5 

4 4 


11 

54 


Portsmouth 

add 

3 4 

10 

6 

4 55 


12 

36p.m. 

Providence 

sub. 

0 41 


7 

6 42 

99 

1 

23 

99 

Albany 

add 

6 34 


8 

6 30 

99 

2 

16 

99 

Amboy 

sub. 

0 39 


9 

7 19 

99 

3 

16 

99 

Sandy Hook 

sub. 

1 8 

6 

10 

8 8 

99 

4 

14 

99 

Philadelphia 

add 

5 15 

6 

11 

8 57 

99 

5 

9 

9 * 

Cape Henlopen 

add 

0 35 

5 

12 

9 46 


6 

0 

99 

Baltimore 

sub. 

4 14 

12 

13 

10 34 

99 

6 

47 

99 

Cape Henry 

add 

0 57 

4 

14 

11 23 

99 

7 

29 

99 

Washington 

sub. 

4 8 


15 

12 12a.m. 

8 

7 


Norfolk 

sub. 

0 52 

7 

16 

1 1 

99 

8 

43 


Charleston 

sub. 

0 22 

5 

17 

1 50 

99 

9 

21 

99 

Key West 

add 

1 16 

2 

18 

2 38 

99 

10 

4 


Havana 

add 

1 35 

3 

19 

3 27 

99 

10 

51 


Rio Janeiro 

sub. 

6 35 

6 

20 

4 16 

99 

11 

42 

99 

Buenos Ayres 


0 0 


21 

5 5 

99 

12 

37 a.m. 

Cape Horn 

sub. 

3 57 

9 

22 

5 54 

99 

1 

37 

99 

Valparaiso 

add 

0 "5 

5 

23 

6 42 

99 

2 

39 

99 

San Francisco 

add 

2 23 

6 

24 

7 31 

99 

3 

42 

99 

Liverpool 

add 

2 39 

25 

25 

8 20 

99 

4 

43 


London 

sub. 

6 30 

18 

26 

9 9 

99 

5 

41 

99 

Hull 

sub. 

2 37 

18 

27 

9 58 

99 

6 

36 

99 

Bremen 

add 

2 37 


28 

10 46 

99 

7 

27 


Lisbon 

sub. 

4 37 


29 

11 35 

99 

8 

14 

99 

Cape Good Hope sub. 

5 37 


29* 

12 

99 

8 

37 

99 

































































Soundings. 


281 


Moon’s Age* High Water* Moon South* 

Example 1. Required the moon’s age on the 25th day of September, 1855? 
Date in September 25d 1 

Epact of September 7d Vadd 

Epact of 1855 lid ISh > 

Reject 30 lod 18h is the moon's age at noon in New York. 

'Example 2. Required the time of high water in New York, and at what time 
the moon is south, on the same date as in preceding example ? 

In the annexed table we have given the moon’s 


Age. South. 

13 days lOh 34m 

Correction add 18hX2= 36m 


High water. 
6h 47 m 
36m 


Moon south at llh lum, and 


Example 3, 

July, 1856. 

Date in July 
Epact of July 
Epact of 1856 
Reject 30 


7h 23m is the time of high 
water in New York 

Find the time of high water in San Francisco on the 18th of 


18d ) 

3d 26h Vadd 
23d lOh ) 


15d 12h the moon’s age. 

In the table Age. High water. 

15 8h 7m 

Correction add 12hX2 24m 

For San Francisco add 2h 23m 


Time of high water in San Francisco loh 54m, July 18, 1856. 

The strength and direction of the wind sometimes accelerates and sometimes 
retards the tide, in consequence the most careful and scientific calculations 
may differ anhour from the time of high water. 


-* •*- 


SOUNDINGS, 


To Reduce Soundings to Low Water. 

Letters denote. 

T — time in hours between high and low water. 

t = time in hours from low water to the time when the soundings are 
taken. 

H= vertical rise of tide in feet from high to low water. 
h — reduction of the sounding taken at the time, t, in feet. 

v = and Ji - 4 E[t + cos. v), 


T 


Example. 


— cos. v when v < 90 
-f cos. v when v > 90 

High water at lOh 15m 
Low water at 3h 45m 


pm. 

5J 


was 16 feet 6 inches. 


Time T — 6h 30m 
The sounding taken at 5h 30m 
Time t = lh 45m 
Vertical rise H = 9'75 feet. 

Required the reduction h — ? and true sounding at low water. 


30 , cos. v — 0*66262. 


_ 180X1-75 = 

6*5 

Reduction h = 1 X 9 - 75 (1—0'66262)=l - 6447 feet. 
Sounding taken at 5h 30m was 16 5 feet. 
Reduction subtract h = 1-6447 

True sounding at low water l4 - 8553 feet. 


24 * 















282 


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Vorj and e describe the letter with the finger in the air. 

For x make a motion up and down with the index finger. 

























283 


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